Question

There are five women and six men in a group. From this group a committee of $4$ is to be chosen. How many different ways can a committee be formed that contain three women and one man?
A. $55$
B. $60$
C. \[25\]
D. \[192\]

Answer 1

Hint: This question will be solved with the help of the topic Permutations and Combinations. Now according to the question, the order of the committee is not mentioned, so we will use combinations and not permutations. Thus, we know that the difference between permutation and combination is that in permutations, the order of the elements is taken into consideration while in combination, the order of elements is not taken into consideration. Now we can define combinations of items as the process in which we can make a selection of a particular number of items from the complete set of items without any repetition.

Complete step by step solution :
According to the question
The total number of women in the group$=5$
Total number of men in the group $=6$
Now we have to form a committee of a total 4 members out of which three are women and one is man .
So using the formula for combination , we get the number of different ways to form committee as
\[{}^{5}{{C}_{3}}\times {}^{6}{{C}_{1}}\]
Now according to the formula of combination , we know that
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
Using this , we get
\[\begin{array}{*{35}{l}}
   {}^{5}{{C}_{3}}=\dfrac{5!}{3!\left( 5-3 \right)!} \\
   ~~~~~=\dfrac{\left( 5\times 4\times 3! \right)}{3!\times 2!} \\
   ~~~~~=\dfrac{5\times 4}{2\times 1} \\
   ~~~~~=5\times 2 \\
   ~~~~~=10 \\
\end{array}\]
Also ,
\[\begin{array}{*{35}{l}}
   {}^{6}{{C}_{1}}=\dfrac{6!}{1!\left( 6-1 \right)!} \\
   ~~~~~~~~~~~~~~~=\dfrac{6\times 5!}{1\times 5!} \\
   ~~~~~~~~~~~~~~~~=\dfrac{6}{1}=6 \\
\end{array}\]
Therefore the number of ways in which the committee can be formed \[={}^{5}{{C}_{3}}\times {}^{6}{{C}_{1}}=10\times 6=60\]
We can form the committee in $60$ different ways such that the committee contains three women and one man.

Therefore, the correct answer is option B.

Note:
If we are given that the committee can be arranged in a particular number of ways only , then it will be permutation . This is because permutation is the way of arranging items in order while combination is the process of selecting from the collection .