A committee of $3$ persons is to be constituted from a group of $2$ men and $3$ women. In how many ways can this be done? How many of these committees would consist of $1$ man and $2$ women?
A committee of $3$ persons is to be constituted from a group of $2$ men and $3$ women. In how many ways can this be done? How many of these committees would consist of $1$ man and $2$ women?
Here, order does not matter. Therefore, we need to count combinations. There will be as many committees as there are combinations of $5$ different persons taken $3$ at a time. Hence, the required number of ways $=\,^{5} C _{3}=\frac{5 !}{3 ! 2 !}=\frac{4 \times 5}{2}=10.$
Now, $1$ man can be selected from $2$ men in $^{2} C _{1}$ ways and $2$ women can be selected from $3$ women in $^{3} C _{2}$ ways. Therefore, the required number of committees
$=\,^{2} C_{1} \times^{3} C_{2}=\frac{2 !}{1 ! 1 !} \times \frac{3 !}{2 ! 1 !}=6$
Similar Questions
A group of $9$ students, $s 1, s 2, \ldots, s 9$, is to be divided to form three teams $X, Y$ and, $Z$ of sizes $2,3$ , and $4$, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is. . . .
A group of $9$ students, $s 1, s 2, \ldots, s 9$, is to be divided to form three teams $X, Y$ and, $Z$ of sizes $2,3$ , and $4$, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is. . . .
- [IIT 2024]
The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by
The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by
A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is
A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is
- [JEE MAIN 2023]
In an election there are $5$ candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote
In an election there are $5$ candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote
The number of ways of dividing $52$ cards amongst four players equally, are
The number of ways of dividing $52$ cards amongst four players equally, are
- [IIT 1979]