A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the…
A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
$196$
$140$
$168$
$176$
Solution
Total number of questions $=13$
Now of questions to be selected $=10$
Restriction - At least 4 from first 5 questions are to be selected.
Case I If exactly 4 out of first 5 questions are selected.
Number of ways of selecting 4 questions from 5 questions $={ }^5 C_4=5$
Remaining $10-4=6$ questions are selected from last $13-5=8$ questions in ${ }^8 C_6$ ways -
$\therefore$ Number of ways $=5 \times{ }^8 C_6=140$
Case II If exactly 5 out of first 5 questions are selected.
Number of ways of selecting 5 out of 5 questions $={ }^5 C_5=1$
Remaining $10-5=5$ questions are selected from last $13-5=8$ questions in ${ }^8 C_5$ ways.
$\therefore$ Number of ways $=1 \times{ }^8 C_5=56$
$\therefore$ Total number of ways $=140+56=196$