Find the numbers of terms in the expansion of the following :
(i) ${(1 + 3 x + 3 x^{2} + x^{3})}^{5}$
(ii) ${(a - b + c)}^{6}$

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Solution

A) $= (1 + 3 x + 3 x^{2} + x^{3})^{5}$
$= ((1 + x)^{3})^{5}$
$= (1 + x)^{15}$
formula for no. terms $= C_{k - 1}^{n + k - 1}$
$= C_{2 - 1}^{15 + 2 - 1}$
$= C_{1}^{16}$
$= 16$
B) $= (a - b + c)^{6}$
formula for no. terms $= C_{k - 1}^{n + k - 1}$
$= C_{3 - 1}^{6 + 3 - 1}$
$= C_{2}^{8}$
$= 28$

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