Question

A college offers 7 courses in the morning and 5 in the evening. Find a possible number of choices with the student who wants to study one course in the morning and one in the evening.
A) $35$
B) $12$
C) $49$
D) $25$

Answer 1

Hint: First find the possible combination of selecting one course in the morning and selecting one course in the evening, then find the possible number of choices with the student who wants to study one course in the morning and one course in the evening by finding the product of both the possible selections of morning and evening.

Complete step-by-step answer:
It is given that the college offers 7 courses in the morning.
So, we can select 1 course in 7 ways.
Therefore, the selection of one course in the morning is given in the form of combination as:
$ = {}^7{C_1}$
Simplify the value of ${}^7{C_1}$getting the possible selection of one course in the morning.
${}^7{C_1} = \dfrac{{7!}}{{1! \times \left( {7 - 1} \right)!}}$
${}^7{C_1} = \dfrac{{7!}}{{6!}} = 7$
So, there are $7$ possible ways to select one course in the morning.
It is also given that the college offers 5 courses in the evening.
It means that we can select 1 course in 5 ways.
Therefore, the selection of one course in the evening is given in the form of combination as:
$ = {}^5{C_1}$
Simplify the value of ${}^5{C_1}$getting the possible selection of one course in the morning.
${}^5{C_1} = \dfrac{{5!}}{{1! \times \left( {5 - 1} \right)!}}$
${}^7{C_1} = \dfrac{{5!}}{{4!}} = 5$
So, there are $5$ possible ways to select one course in the evening.
Therefore, the selection one course in the morning and one in the evening is given as the product of both possible combinations.
$ = $ selection of one course in the morning $ \times $ selection of one course in the evening
$ = {}^7{C_1} \times {}^5{C_1}$
Substitute the value of both the combinations:
$ = 7 \times 5$
$ = 35$
Therefore there are 35 choices with the student to study one course in the morning and one in the evening.
Therefore the option $\left( A \right)$ is correct.

Note: It is given that there are 7 courses available in the morning and we have to choice of these course at one time then the possible number of choices is given as $^7{C_1}$ and similarly, there are five courses available in the evening then the possible number of choosing one course is given as $^5{C_1}$.