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Question
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
\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + \left( \ ^{7}{}{C}_2 + \ ^{7}{}{C}_3 \right) + \left( \ ^{7}{}{C}_3 + \ ^{7}{}{C}_4 \right) + \left( \ ^{7}{}{C}_4 + \ ^{7}{}{C}_5 \right) + \left( \ ^{7}{}{C}_5 + \ ^{7}{}{C}_6 \right) + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\]
\[= 1 + 2 \times \ ^{7}{}{C}_1 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_4 + 2 \times \ ^{7}{}{C}_5 + 2 \times \ ^{7}{}{C}_6 + 1\]
\[= 1 + 2 \times \ ^{7}{}{C}_1 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_6 + 1\]
\[= 2 + 2^2 \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 + \ ^{7}{}{C}_3 \right)\]
\[ = 2 + 2^2 \left( 7 + \frac{7}{2} \times 6 + \frac{7}{3} \times \frac{6}{2} \times 5 \right)\]
\[= 2 + 252 \]
\[ = 254 \]
\[ = 2^8 - 2\]
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