શ્રેણી binom{20}{0} - binom{20}{1} + binom{20 | vedclass

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(D) ધારો કે $S = \binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \dots + \binom{20}{10}$.આપણે જાણીએ છીએ કે $\binom{n}{r} = \binom{n}{n-r}$.તેથી,$\bino

Vedclass · Accepted Answer

$\frac{1}{2} \binom{20}{10}$

Vedclass · Answer

(D) ધારો કે $S = \binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \dots + \binom{20}{10}$.આપણે જાણીએ છીએ કે $\binom{n}{r} = \binom{n}{n-r}$.તેથી,$\binom{20}{0} = \binom{20}{20}$,$\binom{20}{1} = \binom{20}{19}$,...,$\binom{20}{9} = \binom{20}{11}$.$(1-1)^{20} = \sum_{r=0}^{20} (-1)^r \binom{20}{r} = 0$ ધ્યાનમાં લો.આનાથી મળે છે: $\binom{20}{0} - \binom{20}{1} + \dots - \binom{20}{9} + \binom{20}{10} - \binom{20}{11} + \dots + \binom{20}{20} = 0$.સંમિતિનો ઉપયોગ કરતા: $\binom{20}{0} - \binom{20}{1} + \dots - \binom{20}{9} + \binom{20}{10} - \binom{20}{9} + \dots + \binom{20}{0} = 0$.$2[\binom{20}{0} - \binom{20}{1} + \dots - \binom{20}{9}] + \binom{20}{10} = 0$.ધારો કે $X = \binom{20}{0} - \binom{20}{1} + \dots - \binom{20}{9}$. તો $2X + \binom{20}{10} = 0$,તેથી $X = -\frac{1}{2} \binom{20}{10}$.માગેલ સરવાળો $S = X + \binom{20}{10} = -\frac{1}{2} \binom{20}{10} + \binom{20}{10} = \frac{1}{2} \binom{20}{10}$ થાય.

શ્રેણી $\binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \binom{20}{3} + \dots + \binom{20}{10}$ નો સરવાળો શું થાય?

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