The value of the sum ( nC1)2+( nC2)2+( nC3)2+...+( nCn)2 is

Question

The value of the sum ( nC1)2+( nC2)2+( nC3)2+...+( nCn)2 is (A) ( 2nCn)2 (B)  2nCn (C)  2nCn+1 (D)  2nCn-1

Tardigrade Admin · Accepted Answer

We know that (1+ x ) n = n C 0+ n C 1 x + n C 2 x 2+⋅s+ n C n x n ldots ldots text (i) and (x+1)n= n C0 xn+ n C1 xn-1+ n C2 xn-2+⋅s+ n Cn On multiplying equations (i) and (ii), we get beginarrayl (1+x)2 n=( n C0+ n C1 x+ n C2 x2+⋅s+ n Cn xn) × ( n C0 xn+ n C1 xn-1+ n C2 xn-2+⋅s+ n Cn) endarray Coefficient of x n in right hand side =( n C 0)2+( n C 1)2+⋅s+( n C n )2 and beginarrayl text coefficient of x n text in left hand side = 2 n C n ∴ ( n C 0)2+( n C 1)2+⋅s+( n C n )2=(2 n !/ n ! n !) ⇒ ( n C 1)2+⋅s+( n C n )2=((2 n ) !/ n ! n !)-1= 2 n C n -1 endarray

Q. The value of the sum $(^{n} C_{1})^{2} + (^{n} C_{2})^{2} + (^{n} C_{3})^{2} + ... + (^{n} C_{n})^{2}$ is

$(^{2 n} C_{n})^{2}$

$^{2 n} C_{n}$

$^{2 n} C_{n} + 1$

$^{2 n} C_{n} - 1$

Q. The value of the sum (nC1​)2+(nC2​)2+(nC3​)2+...+(nCn​)2 is

(2nCn​)2

2nCn​

2nCn​+1

2nCn​−1

Q. The value of the sum $(^{n} C_{1})^{2} + (^{n} C_{2})^{2} + (^{n} C_{3})^{2} + ... + (^{n} C_{n})^{2}$ is

$(^{2 n} C_{n})^{2}$

$^{2 n} C_{n}$

$^{2 n} C_{n} + 1$

$^{2 n} C_{n} - 1$