(A) The total number of persons is $2 + 3 = 5$. We need to select $3$ persons out of $5$. Since the order does not matter,we use combinations.Total wa

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Total ways: $10$,Committees with $1$ man and $2$ women: $6$

(A) The total number of persons is $2 + 3 = 5$. We need to select $3$ persons out of $5$. Since the order does not matter,we use combinations.Total ways $= ^{5}C_{3} = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10$.To form a committee with $1$ man and $2$ women:We select $1$ man from $2$ men in $^{2}C_{1}$ ways.We select $2$ women from $3$ women in $^{3}C_{2}$ ways.Number of ways $= ^{2}C_{1} \times ^{3}C_{2} = 2 \times 3 = 6$.

Class 11 Mathematics Permutation and Combination Definition of combinations, Condition combinations

Medium

$A$ 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
Total ways: $10$,Committees with $1$ man and $2$ women: $6$
B
Total ways: $10$,Committees with $1$ man and $2$ women: $5$
C
Total ways: $8$,Committees with $1$ man and $2$ women: $6$
D
Total ways: $12$,Committees with $1$ man and $2$ women: $4$

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Class 11

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Permutation and Combination

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Definition of combinations, Condition combinations

There are $10$ persons named $A, B, \dots, J$. We have space to keep only $5$ of them. If $A$ must be included,and $G$ and $H$ must not be included in the group of $5$,in how many ways can we arrange the group in a row?

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$^{20}C_1 + 3 ^{20}C_2 + 3 ^{20}C_3 + ^{20}C_4$ is equal to-

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In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ specific courses are compulsory for every student?

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Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is:

DifficultJEE MAIN 2023

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If a person has $3$ coins of different denominations,the number of different sums that can be formed is

EasyAP EAMCET 2021

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$A$ 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?

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There are $10$ persons named $A, B, \dots, J$. We have space to keep only $5$ of them. If $A$ must be included,and $G$ and $H$ must not be included in the group of $5$,in how many ways can we arrange the group in a row?

$^{20}C_1 + 3 ^{20}C_2 + 3 ^{20}C_3 + ^{20}C_4$ is equal to-

In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ specific courses are compulsory for every student?

Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is:

If a person has $3$ coins of different denominations,the number of different sums that can be formed is

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