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 C0+ n C1 x+ n C2 x2 + ldots+ n Cn xn ...(i) and (x+1)n= n C0 xn+ n C1 xn-1 + n C2 xn-2+ ldots+ n Cn ...(ii) On multiplying Eqs. (i) and (ii), we get (1+x)2 n=( n C0+ n C1 x+ n C2 x2+ ldots+ n Cn xn) ×( n C0 xn+ n C1 xn-1+ n C2 xn-2+ ldots+ n Cn) Coefficient of xn in RHS =( n C0)2+( n C1)2+ ldots+( n Cn)2 and coefficient of xn in LHS = 2 n Cn ∴ ( n C0)2+( n C1)2+ ldots+( n Cn)2=(2 n !/n ! n !) ⇒ ( n C1)2+ ldots+( n Cn)2=((2 n) !/n ! n !)-1 = 2 n Cn-1

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

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

$^{2 n} C_{n}$

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

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

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

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

$^{2 n} C_{n}$

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

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