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20 ∑ r. ( 20 Cr) 2 = ? r=0 Please Give Full explanation

20
∑ r. (20Cr)2 = ?
r=0
 
Please Give Full explanation 

Grade:11

1 Answers

Aditya Gupta
2081 Points
4 years ago
note that we have the following property of summation: ∑r=m to n f(r) = ∑r=m to n f(m+n – r)
so, let S= ∑ r. (20Cr)2 .....(1)
using property S= ∑ (0+20 – r). (20C(0+20 – r))2 
but nCr= nC(n-r)
so, S= ∑ (20 – r). (20Cr)2 .........(2)
add (1) and (2)
2S= ∑ 20(20Cr)2 
S= 10 ∑ (20Cr)2 
now a pretty common result is nc0² + nC1² + nC2² +.......+ nCn² = 2nCn (quite easy to prove, lemme know if you need help).
so, we have S= 10*(40C20)
KINDLY APPROVE :))

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