The value of ( C 1 21 - C 1 10 ) + ( C 2 21 - C 2 10 ) + ( C 3 21 - C 3 10 ) + ( C 4 21 - C 4 10 ) + ? + ( C 10 21 - C 10 10 ) is

Question

The value of $({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + ({}^{21}C_{3} - {}^{10}C_{3}) + ({}^{21}C_{4} - {}^{10}C_{4}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$ is

Smart Achievers · Accepted Answer

(3)

$({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$

$= ({}^{21}C_{1} + {}^{21}C_{2} + \dots + {}^{21}C_{10}) - ({}^{10}C_{1} + {}^{10}C_{2} + \dots + {}^{10}C_{10})$

$= \frac{2^{21}}{2} - 2^{10} = 2^{20} - 2^{10}$

Solution

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

1 $2^{21} - 2^{10}$

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

3 $2^{20} - 2^{10}$

4 $2^{21} - 2^{11}$

Ans.
(3)

$({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$

$= ({}^{21}C_{1} + {}^{21}C_{2} + \dots + {}^{21}C_{10}) - ({}^{10}C_{1} + {}^{10}C_{2} + \dots + {}^{10}C_{10})$

$= \frac{2^{21}}{2} - 2^{10} = 2^{20} - 2^{10}$

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Solution

Q. The value of (C121-C110)+(C221-C210)+(C321-C310)+(C421-C410)+⋯+(C1021-C1010) is

1 221-210

2 220-29

3 220-210

4 221-211

Ans. (3) (C121-C110)+(C221-C210)+⋯+(C1021-C1010) =(C121+C221+⋯+C1021)-(C110+C210+⋯+C1010) =2212-210=220-210