In how many ways can Rs. 16 be divided into 4 person when none of th

Question

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

Options:

A . 70

B . 35

C . 64

D . 192

Answer: Option B
:
B
Required number of ways
= coefficient of

x^{16}

(x^{3} + x^{4} + x^{5} + . . . . . . . x^{7})^{4}

= coefficient of

x^{16}

x^{12} (1 + x + x^{2} + . . . . . . . x^{4})^{4}

= coefficient of

x^{16}

x^{12} (1 - x^{5})^{4} (1 - x)^{- 4}

= coefficient of

x^{4}

(1 - x^{5})^{4} (1 - x)^{- 4}

= coefficient of

x^{4}

(1 - 4 x^{5} + . . . . .)

[1 + 4 x + . . . . + \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

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