First slide

Binomial theorem for positive integral Index

Question

The sixth term in the expansion of $(x^{1 / 3} + {y^{1 / 2})}^{n}$ if the binomial coefficient of the third term form the end is 45, is

Moderate

A

$252 x^{5 / 3} y^{5 / 2}$
B

45
C

$45 x^{5 / 3} y^{5 / 2}$
D

None of these

Solution

Binomial coefficients of the third term from the end in the expansion of
${(x^{1 / 3} + y^{1 / 2})}^{n}$ , is
$^{n} C_{2} = 45 \Rightarrow n (n - 1) - 90 = 0 \Rightarrow n = 10$
$∴$ Sixth term in the expansion of ${(x^{1 / 3} + y^{1 / 2})}^{n}$ is
$^{n} C_{5} {(x^{1 / 3})}^{n - 5} {(y^{1 / 2})}^{5} = 252 x^{5 / 3} y^{5 / 2}$

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