// #define _GLIBCXX_DEBUG #include using namespace std; using uint = unsigned int; using ll = long long; #define CIN( LL , A ) LL A; cin >> A #define GETLINE( A ) string A; getline( cin , A ) #define GETLINE_SEPARATE( A , SEPARATOR ) string A; getline( cin , A , SEPARATOR ) #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr ) #define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( ll VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ ) #define FOREQ( VAR , INITIAL , FINAL ) for( ll VAR = INITIAL ; VAR <= FINAL ; VAR ++ ) #define FOR_ITR( ARRAY , ITR , END ) for( auto ITR = ARRAY .begin() , END = ARRAY .end() ; ITR != END ; ITR ++ ) #define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES ) #define QUIT return 0 #define RETURN( ANSWER ) cout << ( ANSWER ) << "\n"; QUIT #define DOUBLE( PRECISION , ANSWER ) cout << fixed << setprecision( PRECISION ) << ( ANSWER ) << "\n"; QUIT #define MIN( A , B ) A < B ? A : B; #define MAX( A , B ) A < B ? B : A; template inline T Absolute( const T& a ){ return a > 0 ? a : - a; } template using VLArray = list; // 以下、自分のライブラリ（https://github.com/p-adic/cpp）よりソースコードをコピーして編集している。 template INT Residue( const INT& M , const INT& n ) noexcept; template INT Residue( const INT& M , const INT& n ) noexcept { if( M == 0 ){ return 0; } const INT M_abs = ( M > 0 ? M : -M ); if( n < 0 ){ const INT n_abs = -n; const INT res = n_abs % M_abs; return res == 0 ? res : M_abs - res; } return n % M_abs; } using INT_TYPE_FOR_ADIC_INT = long long int; template class AdicInt { private: VLArray m_expansion; INT_TYPE_FOR_ADIC_INT m_n; public: inline AdicInt( const INT_TYPE_FOR_ADIC_INT& n ) noexcept; inline const VLArray& GetExpansion() const noexcept; inline const INT_TYPE_FOR_ADIC_INT& GetValue() const noexcept; static const VLArray& Expand( const INT_TYPE_FOR_ADIC_INT& n ) noexcept; }; template inline AdicInt::AdicInt( const INT_TYPE_FOR_ADIC_INT& n ) noexcept : m_expansion( Expand( n ) ) , m_n( n ) {} template inline const VLArray& AdicInt::GetExpansion() const noexcept { return m_expansion; } template inline const INT_TYPE_FOR_ADIC_INT& AdicInt::GetValue() const noexcept { return m_n; } template const VLArray& AdicInt::Expand( const INT_TYPE_FOR_ADIC_INT& n ) noexcept { static VLArray memory_n{}; static VLArray > memory_answer{}; if( P == 0 ){ // ダミー return memory_n; } auto itr_n = memory_n.begin() , end_n = memory_n.end(); auto itr_answer = memory_answer.begin(); while( itr_n != end_n ){ if( *itr_n == n ){ return *itr_answer; } itr_n++; itr_answer++; } INT_TYPE_FOR_ADIC_INT n_copy = n; VLArray answer{}; if( LENGTH == 0 ){ for( INT_TYPE_FOR_ADIC_INT i = 0 ; n_copy != 0 ; i++ ){ const INT_TYPE_FOR_ADIC_INT d = Residue( P , n_copy ); answer.push_back( d ); n_copy -= d; n_copy /= P; } } else { for( INT_TYPE_FOR_ADIC_INT i = 0 ; i < LENGTH && n_copy != 0 ; i++ ){ const INT_TYPE_FOR_ADIC_INT d = Residue( P , n_copy ); answer.push_back( d ); n_copy -= d; n_copy /= P; } } memory_n.push_back( n ); memory_answer.push_back( answer ); return memory_answer.back(); } // init * ( t ^ num ) template T Power( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true , const string& method = "normal" ); template inline T PowerNormalMethod( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true ); template T PowerBinaryMethod( const T& t , const UINT& num , const T& init = 1 , const bool& for_right_multiplication = true ); // 単なる２乗だが、T次第ではオーバーロードしてより高速なものに置き換える template inline T Square( const T& t ); // PowerBinaryMetodの呼び出しは部分特殊化ではなくオーバーロードで処理できるようにするためにPowerBinaryMethodとはしない。 template inline T Power( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication , const string& method ) { return method == "binary" ? PowerBinaryMethod( t , num , init , for_right_multiplication ) : PowerNormalMethod( t , num , init , for_right_multiplication ); } template inline T PowerNormalMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication ) { return num == 0 ? init : ( for_right_multiplication ? PowerNormalMethod( t , num - 1 , init ) * t : t * PowerNormalMethod( t , num - 1 , init ) ); } template T PowerBinaryMethod( const T& t , const UINT& num , const T& init , const bool& for_right_multiplication ) { const VLArray& num_binary = AdicInt<2>::Expand( num ); T answer = init; T power = t; for( auto itr = num_binary.begin() , end = num_binary.end() ; itr != end ; itr++ ){ if( *itr == 1 ){ answer = for_right_multiplication ? answer * power : power * answer; } // 部分特殊化ではなくオーバーロードで処理できるようにするためにSquareとしない。 power = Square( power ); } return answer; } template inline T Square( const T& t ) { return t * t; } using INT_TYPE_FOR_MOD = long long int; // ここをtempate などにしてしまうとoperator+などを呼び出す際に型推論に失敗する。整数型を変えたい時はINT_TYPE_FOR_MODの型エイリアスを変更する。 template class Mod { protected: INT_TYPE_FOR_MOD m_n; INT_TYPE_FOR_MOD m_inv; public: inline Mod() noexcept; inline Mod( const INT_TYPE_FOR_MOD& n ) noexcept; inline Mod( const Mod& n ) noexcept; inline Mod& operator=( const INT_TYPE_FOR_MOD& n ) noexcept; Mod& operator=( const Mod& n ) noexcept; Mod& operator+=( const INT_TYPE_FOR_MOD& n ) noexcept; inline Mod& operator+=( const Mod& n ) noexcept; inline Mod& operator-=( const INT_TYPE_FOR_MOD& n ) noexcept; inline Mod& operator-=( const Mod& n ) noexcept; Mod& operator*=( const INT_TYPE_FOR_MOD& n ) noexcept; Mod& operator*=( const Mod& n ) noexcept; // INT_TYPE_FOR_MODでの割り算ではないことに注意 virtual Mod& operator/=( const INT_TYPE_FOR_MOD& n ); virtual Mod& operator/=( const Mod& n ); Mod& operator%=( const INT_TYPE_FOR_MOD& n ); inline Mod& operator%=( const Mod& n ); inline Mod operator-() const noexcept; // 前置++/--を使うつもりがないので後置++/--と同じものとして定義する inline Mod& operator++() noexcept; inline Mod& operator++( int ) noexcept; inline Mod& operator--() noexcept; inline Mod& operator--( int ) noexcept; inline const INT_TYPE_FOR_MOD& Represent() const noexcept; void Invert() noexcept; bool CheckInvertible() noexcept; bool IsSmallerThan( const INT_TYPE_FOR_MOD& n ) const noexcept; bool IsBiggerThan( const INT_TYPE_FOR_MOD& n ) const noexcept; }; template inline bool operator==( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator==( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator==( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator==( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator!=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator!=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator!=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator!=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator<( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator<( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator<( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator<=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator<=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator<=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator<=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator>( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator>( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template inline bool operator>=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template inline bool operator>=( const Mod& n0 , const Mod& n1 ) noexcept; template inline bool operator>=( const Mod& n0 , const Mod& n1 ) noexcept; template Mod operator+( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template Mod operator+( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template Mod operator+( const Mod& n0 , const Mod& n1 ) noexcept; template inline Mod operator-( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template Mod operator-( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template Mod operator-( const Mod& n0 , const Mod& n1 ) noexcept; template Mod operator*( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept; template Mod operator*( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept; template Mod operator*( const Mod& n0 , const Mod& n1 ) noexcept; template Mod operator/( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ); template Mod operator/( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ); template Mod operator/( const Mod& n0 , const Mod& n1 ); template Mod operator%( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ); template inline Mod operator%( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ); template inline Mod operator%( const Mod& n0 , const Mod& n1 ); template Mod Inverse( const Mod& n ); template Mod Power( const Mod& n , const INT_TYPE_FOR_MOD& p , const string& method = "normal" ); template <> inline Mod<2> Power( const Mod<2>& n , const INT_TYPE_FOR_MOD& p , const string& method ); // M乗が1になるよう定義されていることに注意 template inline Mod Power( const Mod& n , const Mod& p , const string& method = "normal" ); template <> inline Mod<2> Power( const Mod<2>& n , const Mod<2>& p , const string& method ); // ../Power/a_Body.hppにて定義 template inline T Square( const T& t ); template <> inline Mod<2> Square >( const Mod<2>& t ); template inline string to_string( const Mod& n ) noexcept; template inline basic_ostream& operator<<( basic_ostream& os , const Mod& n ); void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m ); template inline Mod::Mod() noexcept : m_n( 0 ) , m_inv( M ){} template inline Mod::Mod( const INT_TYPE_FOR_MOD& n ) noexcept : m_n( Residue( M , n ) ) , m_inv( 0 ){} template inline Mod::Mod( const Mod& n ) noexcept : m_n( n.m_n ) , m_inv( 0 ){} template inline Mod& Mod::operator=( const INT_TYPE_FOR_MOD& n ) noexcept { return operator=( Mod( n ) ); } template Mod& Mod::operator=( const Mod& n ) noexcept { m_n = n.m_n; m_inv = n.m_inv; return *this; } template Mod& Mod::operator+=( const INT_TYPE_FOR_MOD& n ) noexcept { m_n = Residue( M , m_n + n ); m_inv = 0; return *this; } template inline Mod& Mod::operator+=( const Mod& n ) noexcept { return operator+=( n.m_n ); }; template inline Mod& Mod::operator-=( const INT_TYPE_FOR_MOD& n ) noexcept { return operator+=( -n ); } template inline Mod& Mod::operator-=( const Mod& n ) noexcept { return operator-=( n.m_n ); } template Mod& Mod::operator*=( const INT_TYPE_FOR_MOD& n ) noexcept { m_n = Residue( M , m_n * n ); m_inv = 0; return *this; } template Mod& Mod::operator*=( const Mod& n ) noexcept { m_n = Residue( M , m_n * n.m_n ); if( m_inv == 0 || n.m_inv == 0 ){ m_inv = 0; } else if( m_inv == M || n.m_inv == M ){ m_inv = M; } else { Residue( M , m_inv * n.m_inv ); } return *this; } // 仮想関数なのでinline指定しない。 template Mod& Mod::operator/=( const INT_TYPE_FOR_MOD& n ) { return operator/=( Mod( n ) ); } template Mod& Mod::operator/=( const Mod& n ) { return operator*=( Inverse( n ) ); } template Mod& Mod::operator%=( const INT_TYPE_FOR_MOD& n ) { m_n %= Residue( M , n ); m_inv = 0; return *this; } template inline Mod& Mod::operator%=( const Mod& n ) { return operator%=( n.m_n ); } template inline Mod Mod::operator-() const noexcept { return Mod( 0 ).operator-=( *this ); } template inline Mod& Mod::operator++() noexcept { return operator+=( 1 ); } template inline Mod& Mod::operator++( int ) noexcept { return operator++(); } template inline Mod& Mod::operator--() noexcept { return operator-=( 1 ); } template inline Mod& Mod::operator--( int ) noexcept { return operator-=(); } template inline const INT_TYPE_FOR_MOD& Mod::Represent() const noexcept { return m_n; } template void Mod::Invert() noexcept { if( CheckInvertible() ){ INT_TYPE_FOR_MOD i = m_inv; m_inv = m_n; m_n = i; } else { // m_nがMになるのはここの処理に限るのでRepresent()の値を参照することで例外処理可能 m_n = M; m_inv = M; } return; } template bool Mod::CheckInvertible() noexcept { if( m_inv == 0 ){ LazyEvaluationOfModularInverse( M , m_n , m_inv ); } return m_inv != M; } template inline bool Mod::IsSmallerThan( const INT_TYPE_FOR_MOD& n ) const noexcept { return m_n < Residue( M , n ); } template inline bool Mod::IsBiggerThan( const INT_TYPE_FOR_MOD& n ) const noexcept { return m_n > Residue( M , n ); } template inline bool operator==( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0 == Mod( n1 ); } template inline bool operator==( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return Mod( n0 ) == n0; } template inline bool operator==( const Mod& n0 , const Mod& n1 ) noexcept { return n0.Represent() == n1.Represent(); } template inline bool operator!=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n0 == n1 ); } template inline bool operator!=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return !( n0 == n1 ); } template inline bool operator!=( const Mod& n0 , const Mod& n1 ) noexcept { return !( n0 == n1 ); } template inline bool operator<( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0.IsSmallerThan( n1 ); } template inline bool operator<( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1.IsBiggerThan( n0 ); } template inline bool operator<( const Mod& n0 , const Mod& n1 ) noexcept { return n0.Represent() < n1.Represent(); } template inline bool operator<=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n1 < n0 ); } template inline bool operator<=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return !( n1 < n0 ); } template inline bool operator<=( const Mod& n0 , const Mod& n1 ) noexcept { return !( n1 < n0 ); } template inline bool operator>( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n1 < n0; } template inline bool operator>( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1 < n0; } template inline bool operator>( const Mod& n0 , const Mod& n1 ) noexcept { return n1 < n0; } template inline bool operator>=( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return !( n0 < n1 ); } template inline bool operator>=( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return !( n0 < n1 ); } template inline bool operator>=( const Mod& n0 , const Mod& n1 ) noexcept { return !( n0 < n1 ); } template Mod operator+( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { auto n = n0; n += n1; return n; } template inline Mod operator+( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1 + n0; } template inline Mod operator+( const Mod& n0 , const Mod& n1 ) noexcept { return n0 + n1.Represent(); } template inline Mod operator-( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { return n0 + ( -n1 ); } template inline Mod operator-( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return Mod( n0 - n1.Represent() ); } template inline Mod operator-( const Mod& n0 , const Mod& n1 ) noexcept { return n0 - n1.Represent(); } template Mod operator*( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) noexcept { auto n = n0; n *= n1; return n; } template inline Mod operator*( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) noexcept { return n1 * n0; } template Mod operator*( const Mod& n0 , const Mod& n1 ) noexcept { auto n = n0; n *= n1; return n; } template inline Mod operator/( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) { return n0 / Mod( n1 ); } template inline Mod operator/( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) { return Mod( n0 ) / n1; } template Mod operator/( const Mod& n0 , const Mod& n1 ) { auto n = n0; n /= n1; return n; } template Mod operator%( const Mod& n0 , const INT_TYPE_FOR_MOD& n1 ) { auto n = n0; n %= n1; return n; } template inline Mod operator%( const INT_TYPE_FOR_MOD& n0 , const Mod& n1 ) { return Mod( n0 ) % n1.Represent(); } template inline Mod operator%( const Mod& n0 , const Mod& n1 ) { return n0 % n1.Represent(); } template Mod Inverse( const Mod& n ) { auto n_copy = n; n_copy.Invert(); return n_copy; } template Mod Power( const Mod& n , const INT_TYPE_FOR_MOD& p , const string& method ) { if( p >= 0 ){ return Power,INT_TYPE_FOR_MOD>( n , p , 1 , true , true , method ); } return Inverse( Power( n , -p , method ) ); } template <> inline Mod<2> Power( const Mod<2>& n , const INT_TYPE_FOR_MOD& p , const string& method ) { return p == 0 ? 1 : n; } template inline Mod Power( const Mod& n , const Mod& p , const string& method ) { return Power,INT_TYPE_FOR_MOD>( n , p.Represent() , method ); } template <> inline Mod<2> Power( const Mod<2>& n , const Mod<2>& p , const string& method ) { return p == 0 ? 1 : n; } template <> inline Mod<2> Square >( const Mod<2>& t ) { return t; } template inline string to_string( const Mod& n ) noexcept { return to_string( n.Represent() ) + " + MZ"; } void LazyEvaluationOfModularInverse( const INT_TYPE_FOR_MOD& M , const INT_TYPE_FOR_MOD& n , INT_TYPE_FOR_MOD& m ) { static VLArray memory_M{}; // vectorでなくVLArrayだと引数が小さい順に呼び出した時の計算量がO(1)からO(n)に跳ね上がってしまう。 static VLArray > memory_inverse{}; auto itr_M = memory_M.begin() , end_M = memory_M.end(); auto itr_inverse = memory_inverse.begin(); vector* p_inverse = nullptr; while( itr_M != end_M && p_inverse == nullptr ){ if( *itr_M == M ){ p_inverse = &( *itr_inverse ); } itr_M++; itr_inverse++; } if( p_inverse == nullptr ){ memory_M.push_front( M ); memory_inverse.push_front( vector() ); p_inverse = &( memory_inverse.front() ); p_inverse->push_back( M ); } const INT_TYPE_FOR_MOD size = p_inverse->size(); for( INT_TYPE_FOR_MOD i = size ; i <= n ; i++ ){ p_inverse->push_back( 0 ); } INT_TYPE_FOR_MOD& n_inv = ( *p_inverse )[n]; if( n_inv != 0 ){ m = n_inv; return; } const INT_TYPE_FOR_MOD M_abs = M >= 0 ? M : -M; const INT_TYPE_FOR_MOD n_sub = M_abs % n; INT_TYPE_FOR_MOD n_sub_inv = ( *p_inverse )[n_sub]; if( n_sub_inv == 0 ){ LazyEvaluationOfModularInverse( M , n_sub , n_sub_inv ); } if( n_sub_inv != M ){ n_inv = M_abs - ( ( n_sub_inv * ( M_abs / n ) ) % M_abs ); m = n_inv; return; } for( INT_TYPE_FOR_MOD i = 1 ; i < M_abs ; i++ ){ if( ( n * i ) % M_abs == 1 ){ n_inv = i; m = n_inv; return; } } n_inv = M; m = n_inv; return; } template inline basic_ostream& operator<<( basic_ostream& os , const Mod& n ) { return os << n.Represent(); } constexpr const ll P = 998244353; using MOD = Mod

; int main() { CIN( ll , N ); if( N == 1 ){ RETURN( 0 ); } CIN( ll , K ); CIN( ll , X ); // 条件を満たす各順列PとN以下の正整数iに対し、 // 長さNの順列であってPとi-1項目までは一致しi項目が一致せずPより辞書式順序で小さいものの個数をf(P,i)と置けば // sum_P f(P) = sum_P sum_i f(P,i) = sum_i sum_P f(P,i)となる。 // 従って各iに対しsum_P f(P,i)を計算し総和を取ればよい。 // 各i < Kに対する和 // = sum_P (P[i]未満かつ第i未満成分と一致しないN以下の正整数の個数) * Factorial( N - i ) // = Factorial( N - i ) * sum_P (P[i]未満かつ第i未満成分と一致しないN以下の正整数の個数) // = Factorial( N - i ) * sum( ll j = 0 ; j <= N - i ) ; j++ ) // { j * (条件を満たす順列PのうちP[i]未満かつ第i未満成分と一致しないN以下の正整数の個数がj個であるものの個数) } // = Factorial( N - i ) * sum( ll j = 0 ; j <= N - i ) ; j++ ) // { // j * sum( ll Pi = j + 1 ; Pi <= i + j ; Pi++ ) // { // ( Pi == X ? 0 : Pi未満の正整数をj個選ぶ選び方の個数) // * ( Piより大きいN以下の正整数を(i-1)-((Pi-1)-j)個選ぶ選び方の個数) // * (長さi-1の順列の個数) * (長さN-i-1の順列の個数) // } // } // = Factorial( N - i ) * sum( ll j = 0 ; j <= N - i ) ; j++ ) // { // j * sum( ll Pi = j + 1 ; Pi <= i + j ; Pi++ ) // { // ( Pi == X ? 0 : Pi未満の正整数をj個選ぶ選び方の個数) // * ( Piより大きいN以下の正整数をN-i-j個選ぶ選び方の個数) // * Factrial( i - 1 ) * Factorial( N - i - 1 ) // } // } // = Factorial( N - i ) * Factrial( i - 1 ) * Factorial( N - i - 1 ) // * sum( ll j = 0 ; j <= N - i ) ; j++ ) // { // j * (N以下の正整数をN-i+1個選ぶ選び方のうちXを選びかつj+1番目に小さい数がXでないものの個数) // } // = Factorial( N - i ) * Factrial( i - 1 ) * Factorial( N - i + 1 ) // * sum( ll j = 0 ; j <= N - i ) ; j++ ) // { // j * // ( // (N-1以下の正整数をN-i個選ぶ選び方の個数) // - (X未満の正整数をj個選ぶ選び方の個数) * (Xより大きいN以下の整数をN-i-j個選ぶ選び方の個数) // ) // } // = Factorial( N - i ) * Factrial( i - 1 ) * Factorial( N - i - 1 ) * // ( // sum( ll j = 0 ; j <= N - i ) ; j++ ){ j * Combination( N - 1 , ( N - i ) } // - (Xでない正整数をN-i個選ぶ各選び方に対するX未満の選ばれた数の個数の総和) // ) // = Factorial( N - i ) * Factrial( i - 1 ) * Factorial( N - i - 1 ) * // ( // Combination( N - 1 , i - 1 ) * sum( ll j = 0 ; j <= N - i ) ; j++ ) j // - Combination( N - 1 , N - i ) * ( ( N - i ) * ( X - 1 ) / ( N - 1 ) ) // ) // = Factorial( N - i ) * Factrial( i - 1 ) * Factorial( N - i - 1 ) * // ( // Combination( N - 1 , i - 1 ) * ( ( N - i ) * ( N - i + 1 ) ) / 2 // - Combination( N - 1 , N - i ) * ( ( N - i ) * ( X - 1 ) / ( N - 1 ) ) // ) // = Factorial( N - 1 ) * Factorial( N - i ) * ( ( N - i + 1 ) / 2 - ( X - 1 ) / ( N - 1 ) ) // i = Kに対する和 // = sum_P (X未満かつ第i未満成分と一致しないN以下の正整数の個数) * Factorial( N - K ) // = Factorial( N - K ) * sum( ll j = 0 ; j <= N - K ) ; j++ ) // { // j * (X未満の正整数をj個選ぶ選び方の個数) // * (Xより大きいN以下の正整数を(K-1)-((X-1)-j)個選ぶ選び方の個数) // * (長さK-1の順列の個数) * (長さN-Kの順列の個数) // } // = Factorial( N - K ) * sum( ll j = max( 0 , X - K ) ; j <= min( N - K , X - 1 ) ) ; j++ ) // { // j * Combination( X - 1 , j ) * Combination( N - X , K + j - X ) // * Factorial( K - 1 ) * Factorial( N - K ) // } // = Factorial( N - K ) * Factorial( K - 1 ) * Factorial( N - K ) // * sum( ll j = max( 0 , X - K ) ; j <= min( N - K , X - 1 ) ) ; j++ ) // { j * Combination( X - 1 , j ) * Combination( N - X , N - K - j ) } // = Factorial( N - K ) * Factorial( K - 1 ) * Factorial( N - K - 1 ) // * (Xでない正整数をN-K個選ぶ各選び方に対するX未満の選ばれた数の個数の総和) // = Factorial( N - K ) * Factorial( K - 1 ) * Factorial( N - K ) // * Combination( N - 1 , N - K ) * ( ( N - K ) * ( X - 1 ) / ( N - 1 ) ) // = Factorial( N - 1 ) * Factorial( N - K ) * ( N - K ) * ( X - 1 ) / ( N - 1 ) // i > K全体に対する総和 // = ( Factorial( N - 1 ) / Factorial( N - 1 - ( K - 1 ) ) ) * sum( ll p = 0 ; p < Factorial( N - K ) ; p++ ) p // = ( Factorial( N - 1 ) / Factorial( N - K ) ) * ( Factorial( N - K ) * ( Factorial( N - K ) - 1 ) ) / 2 // = Factorial( N - 1 ) * ( Factorial( N - K ) - 1 ) / 2 MOD answer{}; MOD Factorial[200001]; MOD one{ 1 }; Factorial[0] = one; FOR( i , 1 , N ){ Factorial[i] = Factorial[i-1] * i; } MOD two_inverse = one / 2; MOD rate = MOD( X - 1 ) / ( N - 1 ); ll NK = N - K; FOR( N_minus_i , NK + 1 , N ){ answer += Factorial[N_minus_i] * ( ( N_minus_i + 1 ) * two_inverse - rate ); } answer += Factorial[NK] * NK * rate; answer += ( Factorial[NK] - 1 ) * two_inverse; answer *= Factorial[N-1]; RETURN( answer ); }