A115791 Number of different ways to select n elements from five sets of n elements under the precondition of choosing at least one element from each set.

0, 0, 0, 0, 3125, 97200, 1932805, 31539200, 461828790, 6332578125, 83235183075, 1063505908080, 13327125965725, 164758298214965, 2017489363833125, 24538128923443200, 297028957324770140, 3583456866615114630

Offset: 1

Author

Hieronymus Fischer, Jan 31 2006

Keywords

• nonn

Comments

The number of different ways to select n elements from five sets of n elements under the precondition of choosing at least one element from each set.

Examples

			a(6)=binomial(30,6)-5*binomial(24,6)+10*binomial(18,6)-10*binomial(12,6)+5=97200;
		

Crossrefs

Cf. A115111, A115112, A115246.

Programs

Mathematica

Table[Binomial[5n,n]-5Binomial[4n,n]+10Binomial[3n,n]-10Binomial[2n,n]+5,{n,20}] (* Harvey P. Dale, Nov 06 2011 *)

Formula

a(n) = binomial(5*n,n)-5*binomial(4*n,n)+10*binomial(3*n,n)-10*binomial(2*n,n)+5; ; also: a(n)=sum{binomial(n,i)*binomial(n,j)*binomial(n,k)*binomial(n,l)*binomial(n,m)||i,j,k,l,m=1...(n-4),i+j+k+l+m=n}. General formula for N sets with m elements each: the number of different ways to select k elements from j different sets: G(N,m,j,k) = binomial(N,j)*sum(binomial(j,i)*binomial(i*m,k)*(-1)^i*(-1)^j|i=1...j); Recursion formula: G(N,m,j,k) = binomial(N,j)*binomial(j*m,k) - sum(binomial(N-i,j-i)*G(N,m,i,k)|i=1...j-1);