Find-the-sum-k-0-10-20-C-k-

Question Number 48447 by rahul 19 last updated on 24/Nov/18

F i n d t h e s u m \underset{k = 0}{\sum^{10}}^{20} C_{k} .

Commented by rahul 19 last updated on 24/Nov/18

i t h i n k 2^{n d} l a s t l i n e i s w r o n g \dots

3 t i m e s^{20} C_{10} h a s b e e n t a k e n \dots . .

p l s s a l s o t e l l f i n a l c o r r e c t a n s w e r \dots .

Commented by rahul 19 last updated on 24/Nov/18

Commented by mr W last updated on 24/Nov/18

S = C_{0}^{20} + C_{1}^{20} + C_{2}^{20} + C_{3}^{20} + \dots + C_{10}^{20}

C_{0}^{20} + C_{1}^{20} + C_{2}^{20} + C_{3}^{20} + \dots + C_{10}^{20} + C_{11}^{20} + \dots + C_{20}^{20} = 2^{20}

[(C_{0}^{20} + C_{20}^{20}) + (C_{1}^{20} + C_{19}^{20}) + \dots + (C_{10}^{20} + C_{10}^{20})] - C_{10}^{20} = 2^{20}

[2 C_{0}^{20} + 2 C_{1}^{20} + \dots + 2 C_{10}^{20}] - C_{10}^{20} = 2^{20}

2 [C_{0}^{20} + C_{1}^{20} + \dots + C_{10}^{20}] - C_{10}^{20} = 2^{20}

2 S - C_{10}^{20} = 2^{20}

\Rightarrow S = 2^{19} + \frac{C_{10}^{20}}{2}

Commented by rahul 19 last updated on 24/Nov/18

thanks sir ��