The number of distinct terms in the expansion of ${(x + 2 y - 3 z + 5 w - 7 u)}^{n}$ is

n + 1

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{^{n + 4} C}_{4}

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{^{n + 4} C}_{n}

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\frac{(n + 1) (n + 2) (n + 3) (n + 4)}{24}

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Solution

The correct options are
B ${^{n + 4} C}_{4}$
C ${^{n + 4} C}_{n}$
D $\frac{(n + 1) (n + 2) (n + 3) (n + 4)}{24}$
Consider a binomial expansion
$(x_{1} + x_{2} + x_{3} + x_{4} . . . + x_{m})^{p}$
This will have exactly $^{p + m - 1} C_{p}$ distinct terms.
For the above given expression,
$m = 5$ and $p = n$
Hence the no.of distinct terms will be
$^{n + 5 - 1} C_{n}$
$=^{n + 4} C_{n}$
$=^{n + 4} C_{n + 4 - n}$
$=^{n + 4} C_{4}$
$= \frac{(n + 4) (n + 3) (n + 2) (n + 1)}{4!}$
$= \frac{(n + 4) (n + 3) (n + 2) (n + 1)}{24}$

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