Sum of the series ( 100 C 1)2+2( 100 C 2)2+3( 100 C 3)2+ ldots ldots . .+100( 100 C 100)2 equals

Question

Sum of the series ( 100 C 1)2+2( 100 C 2)2+3( 100 C 3)2+ ldots ldots . .+100( 100 C 100)2 equals (A) (299[1 ⋅ 3 ⋅ 5 ldots ldots . .(199)]/(99) !) (B) 100 ⋅ 200 C 100 (C)  50 ⋅ 200 C 100  (D) 100 ⋅ 199 C 99

Tardigrade Admin · Accepted Answer

S =0 ⋅( 100 C 0)2+1 ⋅( 100 C 1)2+2 ⋅( 100 C 2)2+3 ⋅( 100 C 3)2+ ldots ldots ldots+100 ⋅( 100 C 100)2 S =100 ⋅( 100 C 0)2+99 ⋅( 100 C 1)2+ ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots+0 . ldots ldots ldots+( 100 C 100)2 2 S =100[( 100 C 0)2+( 100 C 1)2+( 100 C 2)2+ ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots ldots+( 100 C 100)2] 2 S =100 ⋅ 200 C 100=100 ⋅ ((200) !/100 ! ⋅ 100 !) ⇒ 2 S =(100 ⋅ 200 ⋅(199) !/100 ⋅(99) ! ⋅(100) !) ⇒ S =100 ⋅ 199 C 99

Q. Sum of the series $(^{100} C_{1})^{2} + 2 (^{100} C_{2})^{2} + 3 (^{100} C_{3})^{2} + \dots\dots .. + 100 (^{100} C_{100})^{2}$ equals

$\frac{2 ^{99} [ 1 \cdot 3 \cdot 5 \dots\dots .. ( 199 )]}{( 99 )!}$

$100 \cdot^{200} C_{100}$

$50 \cdot^{200} C_{100}$

$100 \cdot^{199} C_{99}$

Q. Sum of the series (100C1​)2+2(100C2​)2+3(100C3​)2+……..+100(100C100​)2 equals

(99)!299[1⋅3⋅5……..(199)]​

100⋅200C100​

50⋅200C100​

100⋅199C99​

Q. Sum of the series $(^{100} C_{1})^{2} + 2 (^{100} C_{2})^{2} + 3 (^{100} C_{3})^{2} + \dots\dots .. + 100 (^{100} C_{100})^{2}$ equals

$\frac{2 ^{99} [ 1 \cdot 3 \cdot 5 \dots\dots .. ( 199 )]}{( 99 )!}$

$100 \cdot^{200} C_{100}$

$50 \cdot^{200} C_{100}$

$100 \cdot^{199} C_{99}$