If the coefficients of $(2 r + e)^{t h}$ term and $(3 + 4)^{t h}$ term in the expansion of $(1 + x)^{21}$ are equal, then find $r$ .

Open in App

Solution

Given expression is $(1 + x)^{21}$ .
$(2 r + e)^{t h}$ term is $21_{C_{2 r + e - 1}} 1^{2 r + e - 1} \times x^{21 - (2 r + e - 1)}$
$(3 + 4)^{t h}$ term is $21_{C_{3 + 4 - 1}} 1^{3 + 4 - 1} \times x^{21 - (3 + 4 - 1)}$
$(3 + 4)^{t h}$ term is $21_{C_{6}} \times 1 \times x^{15}$
Coefficient of $(2 r + e)^{t h}$ term is $21_{C_{2 r + e - 1}} \times 1^{2 r + e - 1} = 21_{C_{2 r + e - 1}}$
Coefficient of $(3 + 4)^{t h}$ term is $21_{C_{6}} \times 1^{6} = 21_{C_{6}}$
Now both are equal. so
$21_{C_{2 r + e - 1}} = 21_{C_{6}} n_{C_{k}} = n_{C_{n - k}} \Rightarrow 2 r + e - 1 = k \Rightarrow 6 = 21 - k \Rightarrow 2 r + e - 1 = 21 - 6 \Rightarrow r = \frac{16 - e}{2}$

Suggest Corrections

Similar questions

{(2 r + 4)}^{t h}, {(r - 2)}^{t h}

{(1 + x)}^{18}

If the coefficient of $(2 r + 4)^{t h}$ and $(r - 2)^{t h}$ terms in the expansion of ( $1 + x)^{18}$ are equal, then r =

(2 r + 4)^{t h}

(r - 2)^{t h}

(1 + x)^{18}

r

If the coefficients of (2r+1)th term and (r+2)th terms in the expansion of $(1 + x)^{43}$ are equal, find r.

(2 r + 4)

(r - 2)

(1 + x)^{18}

r =

Join BYJU'S Learning Program