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A student has to answer 10 questions, choosing at least 4 from each of the parts $A$ and $B$. If there are 6 questions in part $A$ and 7 in part $B$, then the number of ways can the student choose 10 questions is
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Verified Answer
The correct answer is:
266
Given,
Total number of questions in part $A=6$
Total number of questions in part $B=7$
Pattern to choose 10 questions from 13 questions is
4 from part $A$ and 6 from part $B$ or
5 from part $A$ and 5 from part $B$ or
6 from part $A$ and 4 from part $B$.
$\therefore$ Total required number of ways
$\begin{aligned}
&=\left({ }^{6} C_{4} \times{ }^{7} C_{6}\right)+\left({ }^{6} C_{5} \times{ }^{7} C_{5}\right)+\left({ }^{6} C_{6} \times{ }^{7} C_{4}\right) \\
&=(15 \times 7)+(6 \times 21)+(1 \times 35) \\
&=105+126+35=266
\end{aligned}$
Total number of questions in part $A=6$
Total number of questions in part $B=7$
Pattern to choose 10 questions from 13 questions is
4 from part $A$ and 6 from part $B$ or
5 from part $A$ and 5 from part $B$ or
6 from part $A$ and 4 from part $B$.
$\therefore$ Total required number of ways
$\begin{aligned}
&=\left({ }^{6} C_{4} \times{ }^{7} C_{6}\right)+\left({ }^{6} C_{5} \times{ }^{7} C_{5}\right)+\left({ }^{6} C_{6} \times{ }^{7} C_{4}\right) \\
&=(15 \times 7)+(6 \times 21)+(1 \times 35) \\
&=105+126+35=266
\end{aligned}$
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