Question

In how many ways can Rs. 16 be divided into 4 person when none of them get less than Rs. 3

• A. 70
• B. 35
• C. 64
• D. 192

Answer 1

The correct option is B

35

Required number of ways

= coefficient of $x^{16}$ in $(x^3+x^4+x^5+.......x^7{)}^4$

= coefficient of $x^{16}$ in $x^{12}(1+x+x^2+.......x^4{)}^4$

= coefficient of $x^{16}$ in $x^{12}(1-x^5{)}^4(1-x{)}^{-4}$

= coefficient of $x^4$ in $(1-x^5{)}^4(1-x{)}^{-4}$

= coefficient of $x^4$ in $(1-4x^5+.....)$

$[1+4x+....+\frac{(r+1)(r+2)(r+3)}{3!}x]$

= $\frac{(4+1)(4+2)(4+3)}{3!}$ = 35

Aliter: Remaining 4 rupees can be distributed in ${}^{4+4-1}{C}_{4-1}$ i.e., 35 ways