The value of the expression ( displaystyle∑r=010 10 C r)( displaystyle∑K=010(-1)K ( 10 C K /2 K )) is :

Question

The value of the expression ( displaystyle∑r=010 10 C r)( displaystyle∑K=010(-1)K ( 10 C K /2 K )) is : (A) 210 (B) 220 (C) 1 (D) 25

Tardigrade Admin · Accepted Answer

( displaystyle∑r=010 10 Cr)( displaystyle∑k=010(-1)k ( 10 Ck/2k)) =( 10 C 0+ ldots ldots . .+ 10 C 10)( 10 C 0-( 10 C 1/2)+( 10 C 2/22) ldots ldots . .+( 10 C 10/210)) =210 ×(1-(1/2))10=1

Q. The value of the expression $(r = 0 \sum 10^{10} C_{r}) (K = 0 \sum 10 (- 1)^{K} \frac{^{10} C _{K}}{2 ^{K}})$ is :

$2^{10}$

$2^{20}$

1

$2^{5}$

$(r = 0 \sum 10^{10} C_{r}) (k = 0 \sum 10 (- 1)^{k} \frac{^{10} C _{k}}{2 ^{k}})$
$= (^{10} C_{0} + \dots\dots .. +^{10} C_{10}) (^{10} C_{0} - \frac{^{10} C _{1}}{2} + \frac{^{10} C _{2}}{2 ^{2}} \dots\dots .. + \frac{^{10} C _{10}}{2 ^{10}})$
$= 2^{10} \times (1 - \frac{1}{2})^{10} = 1$

Q. The value of the expression (r=0∑10​10Cr​)(K=0∑10​(−1)K2K10CK​​) is :

210

220

1

25

(r=0∑10​10Cr​)(k=0∑10​(−1)k2k10Ck​​) =(10C0​+……..+10C10​)(10C0​−210C1​​+2210C2​​……..+21010C10​​) =210×(1−21​)10=1

Q. The value of the expression $(r = 0 \sum 10^{10} C_{r}) (K = 0 \sum 10 (- 1)^{K} \frac{^{10} C _{K}}{2 ^{K}})$ is :

$2^{10}$

$2^{20}$

$2^{5}$

$(r = 0 \sum 10^{10} C_{r}) (k = 0 \sum 10 (- 1)^{k} \frac{^{10} C _{k}}{2 ^{k}})$
$= (^{10} C_{0} + \dots\dots .. +^{10} C_{10}) (^{10} C_{0} - \frac{^{10} C _{1}}{2} + \frac{^{10} C _{2}}{2 ^{2}} \dots\dots .. + \frac{^{10} C _{10}}{2 ^{10}})$
$= 2^{10} \times (1 - \frac{1}{2})^{10} = 1$