The value of $^{13} C_{7} +^{13} C_{8} +^{13} C_{9} +^{13} C_{10} +^{13} C_{11} +^{13} C_{12} +^{13} C_{13}$

2^{12}

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2^{11}

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\frac{2^{13} + 1}{2}

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\frac{2^{13} - 1}{2}

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Solution

The correct option is A $2^{12}$
${13_{C}}_{7} + {13_{C}}_{8} + {13_{C}}_{9} + {13_{C}}_{10} + {13_{C}}_{11} + {13_{C}}_{12} + {13_{C}}_{13}$

$= 2^{13} - {(13_{C}}_{0} + {13_{C}}_{1} + {13_{C}}_{2} + {13_{C}}_{3} + {13_{C}}_{4} + {13_{C}}_{5} + {13_{C}}_{6})$

$A l s o, {(13_{C}}_{0} + {13_{C}}_{1} + {13_{C}}_{2} + {13_{C}}_{3} + {13_{C}}_{4} + {13_{C}}_{5} + {13_{C}}_{6}) = {13_{C}}_{7} + {13_{C}}_{8} + {13_{C}}_{9} + {13_{C}}_{10} + {13_{C}}_{11} + {13_{C}}_{12} + {13_{C}}_{13}$

$T h e r e f o r e {13_{C}}_{7} + {13_{C}}_{8} + {13_{C}}_{9} + {13_{C}}_{10} + {13_{C}}_{11} + {13_{C}}_{12} + {13_{C}}_{13} = \frac{2^{13}}{2} = 2^{12}$

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