A student has to answer 10 questions, choosing at least 4 from each of part A and part B.
Question:

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

Solution:

The various possibilities for answering the 10 questions are given below:

(i) 4 from part A and 6 from part B.

(ii) 5 from part A and 5 from part B.

(iii) 6 from part A and 4 from part B.

$\therefore$ Required number of ways $={ }^{6} C_{4} \times{ }^{7} C_{6}+{ }^{6} C_{5} \times{ }^{7} C_{5}+{ }^{6} C_{6} \times{ }^{7} C_{4}$

$=\frac{6 !}{4 ! 2 !} \times 7+6 \times \frac{7 !}{5 ! 2 !}+1 \times \frac{7 !}{4 ! 3 !}$

$=105+126+35$

= 266