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Question: The greatest coefficient in the expansion of (x + y + z + t)<sup>15</sup> is...

The greatest coefficient in the expansion of (x + y + z + t)15 is

A

15!3!(4!)2\frac{15!}{3!(4!)^{2}}

B

15!3!4!\frac{15!}{3!4!}

C

5!3!(4!)3\frac{5!}{3!(4!)^{3}}

D

15!3!(4!)3\frac{15!}{3!(4!)^{3}}

Answer

15!3!(4!)3\frac{15!}{3!(4!)^{3}}

Explanation

Solution

greatest coefficient = n!(q!)m−r((q+1)!)r\frac{n!}{(q!)^{m - r}((q + 1)!)^{r}}

where q is quotient and r is remainder when n is divided by m.

Here n = 15 and m = 4 q = 3 & r = 3

So coeff. = 15!3!(4!)3\frac{15!}{3!(4!)^{3}}