Question

Find the number of ways in which $12$ identical coins can be distributed in $6$ different purses, if not more than $3$ and not less than $1$ coin goes in each purse ____

Answer 1

Required no. is the same as the no. of ways of choosing $6$ whole numbers $x_{1,}{x}_{2,}{x}_{3,}{x}_{4,}{x}_{5,}{x}_6$ such that$x_1+x_2+x_3+x_4+x_5+x_6=12$.

This in term is equal to the coeff $x^{12}$ in

$(x^0+x^1+x^2+...+x^{12})(x^0+x^1+x^2+...+x^{12})...$ upto $6$ factors,i.e,

coeff of $x^{12}$ in $(1+x+x^2+...+x^{12})=$coeff of $x^{12}$ in ${\left(\frac{1-x^{13}}{1-x}\right)}^6$

$=$ coeff of $x^{12}$ in ${(1-x^{13})}^6{(1-x)}^{-6}$

$=$ coeff of $x^{12}$ in ${(1-x)}^{-6}$=${}^{12+6-1}{C}_{12}$

{$\because$ coeff of $x^n$ in ${(1-x)}^{-k}$,KGN is ${}^{n+k-1}{C}_n$

$={}^{17}{C}_{12}=6188$