Question

A student has to answer $10$ questions, choosing at least $4$ from each of parts $A$ and $B$. If there are $6$ questions in Part $A$ and $7$ in Part $B$, in how many ways can the student choose $10$ questions?

• A. 266
• B. 260
• C. 256
• D. 270

Answer 1

Answer: (A)

The possibilities are:
$4$ 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, the required number of ways is
$={ }^{6} C _{4} \times{ }^{7} C _{6}+{ }^{6} C _{5} \times{ }^{7} C _{5}+{ }^{6} C _{6} \times{ }^{7} C _{4}$
$=105+126+35=266$