First slide

Combinations

Question

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

Moderate

A

70
B

35
C

64
D

192

Solution

Required number of ways
= Coefficient of x¹⁶ in ${(x^{3} + x^{4} + x^{5} + \dots + x^{16})}^{4}$
= Coefficient of x¹⁶ in $x^{12} {(1 + x + x^{2} + \dots + x^{12})}^{4}$
= Coefficient of x⁴ in ${(1 - x^{13})}^{4} (1 - x)^{- 4}$
= Coefficient of x⁴ in $(1 - 13 x^{5} + \dots)$ $\times [1 + 4 x + \dots + \frac{(r + 1) (r + 2) (r + 3)}{3!} x^{r}]$
$= \frac{(4 + 1) (4 + 2) (4 + 3)}{3!} = 35$

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