Find the value of $^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + . . . . . . +^{2 n + 1} C_{n} +^{2 n + 1} C_{n + 1} +^{2 n + 1}$ $C_{n + 2} + . . . . . +^{2 n + 1} C_{2 n + 1}$ if $n = 3$ .
___

Open in App

Solution

We know,
Sum of the coefficients of binomial expansion
$^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + . . . . . . +^{2 n + 1} C_{n} = 2^{n}$
$^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + . . . . . . +^{2 n + 1}$ $C_{n} +^{2 n + 1} C_{n + 1} +^{2 n + 1} C_{n + 2} + . . . . . +^{2 n + 1} C_{2 n + 1} = 2^{2 n + 1}$

Put n=3
$2^{6 + 1} = 2^{7} = 128$

Suggest Corrections

Similar questions

If T= $^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + . . . . . . +^{2 n + 1} C_{n} = 255$ . Find the value of n.

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

\sum \sum 0 \leq r < s \leq n (r + s) C_{r} C_{s}

\frac{1}{2}^{n} C_{1} .^{n - 1} C_{1} + \frac{2}{3}^{n} C_{2} + \frac{3}{4}^{n} C_{c} .^{n - 1} C_{3} + \dots \dots

^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}

x^{n - 1}

(x +^{2 n + 1} C_{0}) (x +^{2 n + 1} C_{1}) (x +^{2 n + 1} C_{2}) \dots (x +^{2 n + 1} C_{n})

Join BYJU'S Learning Program