Question

In how many ways a group of 3 boys and 2 girls can be formed out of a total of 4 boys and 4 girls?
A) 15
B) 16
C) 20
D) 24

Answer 1

Hint:We have to choose 3 boys and 2 girls out of 4 boys and 4 girls so we will apply the formula for combinations for both boys and girls and find each of the values and then multiply them to get the answer.
The formula for combinations is given by:-
\[{}^nCr = \dfrac{{n!}}{{\left( {r!} \right)\left( {n - r} \right)!}}\]
where r elements are to be chosen out of n elements.

Complete step-by-step answer:
We have select 3 boys out of 4 boys therefore,
\[
  n = 4 \\
  r = 3 \\
 \]
Applying the formula of combinations we get:-
\[
  {}^4C3 = \dfrac{{4!}}{{\left( {3!} \right)\left( {4 - 3} \right)!}} \\
  {\text{ }} = \dfrac{{4!}}{{\left( {3!} \right)\left( {1!} \right)}} \\
  {\text{ }} = \dfrac{{4 \times 3!}}{{3! \times 1}} \\
  {\text{ }} = {\text{4}}...............{\text{(1)}} \\
 \]
Now, we have to choose 2 girls out of 4 girls therefore,
\[
  n = 4 \\
  r = 2 \\
 \]
Applying the formula of combinations we get:-
\[
  {}^4C2 = \dfrac{{4!}}{{\left( {2!} \right)\left( {4 - 2} \right)!}} \\
  {\text{ }} = \dfrac{{4!}}{{\left( {2!} \right)\left( {2!} \right)}} \\
  {\text{ }} = \dfrac{{4 \times 3 \times 2!}}{{2! \times 2 \times 1}} \\
  {\text{ }} = \dfrac{{4 \times 3}}{2} \\
  {\text{ }} = 2 \times 3 \\
  {\text{ }} = 6...............{\text{(2)}} \\
 \]
Now the total number of ways in which the group of 3 boys and 2 girls can be formed out of a total of 4 boys and 4 girls is given by:-
\[{\text{Total number of ways = number of ways in which boys are selected }} \times {\text{number of ways in which girls are selected}}\]
Putting in the values from equation 1 and equation 2 we get:-
\[
  {\text{Total number of ways }} = 4 \times 6 \\
  {\text{Total number of ways }} = 24{\text{ ways}} \\
 \]

Hence, the total number of ways in which the group can be formed are 24 ways.

Note:The combination formula is used to find the number of ways of selecting items from a collection, such that the order of selection does not matter.
Factorial of a number is the product of all the positive integers less than or equal to that number and is denoted by \[n!\]