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

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

1. A

70

2. B

35

3. C

64

4. D

192

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Solution:

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

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

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