text If ∑r=025 50 Cr ⋅ 50-r C25-r =K( 50 C25) text , then K text is equal to:

Question

text If ∑r=025 50 Cr ⋅ 50-r C25-r =K( 50 C25) text , then K text is equal to:  (A) 225 - 1 (B) (25)2 (C) 225 (D) 224

Tardigrade Admin · Accepted Answer

∑25r=0 50Cr . 50-rC25-r =∑25r=0 (50!/r!(50-r)!) ×((50-r)!/(25)!(25-r)!) =∑25r=0 (50!/25!25!) ×(25!/(25-r)!(r!)) = 50C25 ∑25r=0 25Cr =(225)50C25 ∴ K = 225

Q. $If \sum_{r = 0}^{25} {^{50} C_{r} \cdot^{50 - r} C_{25 - r}} = K (^{50} C_{25}), then K is equal to :$

$2^{25} - 1$

$(25)^{2}$

$2^{25}$

$2^{24}$

Q. If ∑r=025​{50Cr​⋅50−rC25−r​}=K(50C25​), then K is equal to :

225−1

(25)2

225

224

∑r=025​50Cr​.50−rC25−r​ =∑r=025​r!(50−r)!50!​×(25)!(25−r)!(50−r)!​ =∑r=025​25!25!50!​×(25−r)!(r!)25!​ =50C25​∑r=025​25Cr​=(225)50C25​ ∴K=225

Q. $If \sum_{r = 0}^{25} {^{50} C_{r} \cdot^{50 - r} C_{25 - r}} = K (^{50} C_{25}), then K is equal to :$

$2^{25} - 1$

$(25)^{2}$

$2^{25}$

$2^{24}$