Question

Brenda can choose between 2 pairs of pants and 3 shirts how many outfits are possible?

Answer 1

In this question we have to find the number of outfits can be chosen from the given number of pants and shirts, for this we will the combination formula which is given by Number of combinations when ‘r’ elements are selected out of a total of ‘n’ elements is ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, which can also be represented by ${}^n{C_r} = {}^n{C_{n - r}}$.

Complete step by step solution:
Given question is Brenda can choose between 2 pairs of pants and 3 shirts,
Number of shirts = 3,
Number of pants = 2,
Now we will use the combination formula which is given by Number of combinations when ‘r’ elements are selected out of a total of ‘n’ elements is ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$,
We have to choose 1 shirt from 3 shirts and 1 from 2 pair of pants,
First we will choose shirts, by using the formula, here $n = 3$ and $r = 1$,
Substituting the values in the formula we get,
$\Rightarrow {}^3{C_1} = \dfrac{{3!}}{{1!\left( {3 - 1} \right)!}}$,
Now simplifying we get,
$\Rightarrow {}^3{C_1} = \dfrac{{3!}}{{1!\left( 2 \right)!}}$,
Now simplifying using factorial we get,
$\Rightarrow {}^3{C_1} = \dfrac{{3 \times 2 \times 1}}{{1\left( {2 \times 1} \right)}}$,
Now eliminating the like terms we get,
$\Rightarrow {}^3{C_1} = 3$,
So, the number of ways Brenda can choose a shirt is 3,
Now we will choose a pant by using the formula, here $n = 2$and $r = 1$,
Substituting the values in the formula we get,
$\Rightarrow {}^2{C_1} = \dfrac{{2!}}{{1!\left( {2 - 1} \right)!}}$,
Now simplifying we get,
$\Rightarrow {}^2{C_1} = \dfrac{{2!}}{{1!\left( 1 \right)!}}$,
Now simplifying using factorial we get,
$\Rightarrow {}^2{C_1} = \dfrac{{2 \times 1}}{{1\left( 1 \right)}}$,
Now eliminating the like terms we get,
$\Rightarrow {}^2{C_1} = 2$,
So, the number of ways Brenda can choose a pant is 2,
Now multiply the number of ways choosing a shirt and number of ways choosing a pant to get a total number outfits,
So, number of outfits Brenda can wear$= 3 \times 2 = 6$,
Final Answer:
$\therefore$ The number of ways Brenda can choose the outfits will be equal to 6.

Note:
Combination is the different selections of a given number of elements taken one by one, or some, or all at a time. For example, if we have two elements A and B, then there is only one way to select two items, we select both of them. As the question is related to combinations, we should know the definition and the formula related to the combinations and students should understand the question, and the condition given, as they may get confused in finding the arrangements, which should be done according to the condition given in the question.