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The Value of ( 7 C 0 + 7 C 1 ) + ( 7 C 1 + 7 C 2 ) + . . . + ( 7 C 6 + 7 C 7 ) is (A) 27 − 1 (B) 28 − 2 (C) 28 − 1 (D) 28 - Mathematics

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MCQ

The value of\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + . . . + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\] is

Options

  • 27 − 1

  •  28 − 2

  •  28 − 1

  • 28

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Solution

28 − 2

\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + \left( \ ^{7}{}{C}_2 + \ ^{7}{}{C}_3 \right) + \left( \ ^{7}{}{C}_3 + \ ^{7}{}{C}_4 \right) + \left( \ ^{7}{}{C}_4 + \ ^{7}{}{C}_5 \right) + \left( \ ^{7}{}{C}_5 + \ ^{7}{}{C}_6 \right) + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\]

\[= 1 + 2 \times \ ^{7}{}{C}_1 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_4 + 2 \times \ ^{7}{}{C}_5 + 2 \times \ ^{7}{}{C}_6 + 1\]

\[= 1 + 2 \times \ ^{7}{}{C}_1 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_6 + 1\]
\[= 2 + 2^2 \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 + \ ^{7}{}{C}_3 \right)\]
\[ = 2 + 2^2 \left( 7 + \frac{7}{2} \times 6 + \frac{7}{3} \times \frac{6}{2} \times 5 \right)\]

\[= 2 + 252 \]
\[ = 254 \]
\[ = 2^8 - 2\]

Concept: Combination
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 17 Combinations
Q 23 | Page 26

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