Name: Middle Term in Binomial Expansion: Illustration
Uploaded: 2019-07-19
Duration: 3 min 24 s
Description: We have previously studied what Binomial Theorem is. We have learnt its expansion. Let's now take a look at this video to observe the Middle Term in Binomial...

Question

The binomial coefficient of the third term from the end in the expansion of

{(y^{\frac{2}{3}} + x^{\frac{5}{4}})}^{n}

is

91

. Find the

9^{t h}

term of the expansion.

Accepted Answer

Given, ${(y^{\frac{2}{3}} + x^{\frac{5}{4}})}^{n}$

Number of terms $= n + 1$ .

So, the third term from the end will be $(n + 1) - 3 + 1 = n - 1$

$(r + 1)^{t h}$ term in the expansion of $(a + b)^{n}$ is ${}^{n}C_{r} a^{n - r} b^{r}$ .

$∴ 3^{r d}$ term from the end in the expansion of ${(y^{\frac{2}{3}} + x^{\frac{5}{4}})}^{n} = {}^{n}C_{n - 2} {(y^{\frac{2}{3}})}^{2} {(x^{\frac{5}{4}})}^{n - 2}$

According to given condition, we get ${}^{n}C_{n - 2} = 91$ .

$\Rightarrow \frac{n!}{(n - 2)! 2!} = 91$ $[∵ {}^{n}C_{r} = \frac{n!}{r! (n - r)!}]$

$\Rightarrow \frac{n (n - 1)}{2} = 91$ $[∵ n! = n (n - 1)! = n (n - 1) (n - 2)! = . . .]$

$\Rightarrow n (n - 1) = 182$

$\Rightarrow n (n - 1) = 14 \times 13$

$∴ n = 14$ .

$∴ 9^{t h}$ term in the expansion of ${(y^{\frac{2}{3}} + x^{\frac{5}{4}})}^{n} = {}^{14}C_{8} {(y^{\frac{2}{3}})}^{6} {(x^{\frac{5}{4}})}^{8}$

$= {}^{14}C_{6} y^{4} x^{10}$ $[∵ {}^{n}C_{r} = {}^{n}C_{n - r}]$

$= \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \times y^{4} \times x^{10}$ $[∵ {}^{n}C_{r} = \frac{n!}{r! (n - r)!} = \frac{n (n - 1) (n - 2) \dots (n - r + 1)}{r!}]$

$= 3003 y^{4} x^{10}$ .

The binomial coefficient of the third term from the end in the expansion of ${(y^{\frac{2}{3}} + x^{\frac{5}{4}})}^{n}$ is $91$ . Find the $9^{t h}$ term of the expansion.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

The binomial coefficient of the third term from the end in the expansion of y23+x54n is 91. Find the 9th term of the expansion.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

The binomial coefficient of the third term from the end in the expansion of ${(y^{\frac{2}{3}} + x^{\frac{5}{4}})}^{n}$ is $91$ . Find the $9^{t h}$ term of the expansion.