A women student is to answer 10 out of 13 questions. Find the number of her choices where

Question

she must answer: (a) the first two questions; (b) the first or second question but not both; (c)
exactly 3 out of the first 5 questions; (d) at least 3 of the first 5 questions.

Answer 1

(a) The student must answer the first two questions. There is only 1 choice for this.

(b) The student can choose to answer the first question but not the second, or answer the second question but not the first. This gives 2 choices.

(c) The student must answer exactly 3 out of the first 5 questions. This can be done in $\binom{5}{3}=10$ ways.

(d) The student can choose to answer any 3, 4, or 5 questions out of the first 5. This can be done in $\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=10+5+1=16$ ways.

Therefore:
(a) 1 choice
(b) 2 choices
(c) 10 choices
(d) 16 choices