The value of (21C1 - 10C1) + (21C2 - 10C2) + (21C3 - 10C3) +(21C4 - 10C4) +....+(21C10 - 10C10) is :

Question

The value of (21C1 - 10C1) + (21C2 - 10C2) + (21C3 - 10C3) +(21C4 - 10C4) +....+(21C10 - 10C10) is : (A) 221 - 210 (B) 220 - 29 (C) 220 - 210 (D) 221 - 211

Tardigrade Admin · Accepted Answer

= 21C1+21C2+...+21C10 (1/2) 21C0+21C1+...+21C21 -1 = 220-1 (10C1+10C2+...+10C10) = 210-1 ∴ Required sum = (220 - 1) - (210 - 1) = 220 - 210

Q. The value of
$(^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) + .... + (^{21} C_{10} -^{10} C_{10})$ is :

$2^{21} - 2^{10}$

$2^{20} - 2^{9}$

$2^{20} - 2^{10}$

$2^{21} - 2^{11}$

$=^{21} C_{1} +^{21} C_{2} + ... +^{21} C_{10} \frac{1}{2} {^{21} C_{0} +^{21} C_{1} + ... +^{21} C_{21}} - 1$
$= 2^{20} - 1$
$(^{10} C_{1} +^{10} C_{2} + ... +^{10} C_{10}) = 2^{10} - 1$
$∴$ Required sum $= (2^{20} - 1) - (2^{10} - 1)$
$= 2^{20} - 2^{10}$

Q. The value of (21C1​−10C1​)+(21C2​−10C2​)+(21C3​−10C3​)+(21C4​−10C4​)+....+(21C10​−10C10​) is :

221−210

220−29

220−210

221−211