Question

The number of ways, in which 12 identical coins can be put into 5 different purses, if none of the purses remain empty, is

• A. 660
• B. 110
• C. 165
• D. 330

Answer 1

The correct option is D 330

Coefficient of $x^{12}$ in $(x^1+x^2+x^3+x^4+x^5+...{)}^5$
$=$ coefficient of $x^7$ in $(1+x+x^2+x^3+...{)}^5$
$=$ coefficient of $x^7$ in $(1-x{)}^{-5}$
$=1{+}^5{C}_{1}x{+}^6{C}_2{x}^2+...{+}^{11}{C}_7{x}^7$
$=$ coefficient of ${}^{11}{C}_7=\frac{\mathrm{11.10.9.8}}{\mathrm{4.3.2.1}}=11\times 30=330$