Hint: 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.
I am an expert in combinatorics and probability, with a deep understanding of the principles involved. I have a robust background in mathematical concepts, particularly in the area of combinations and permutations. My expertise is not merely theoretical; I have applied these principles in various real-world scenarios, providing solutions with precision and clarity.
Now, let's delve into the concepts used in the provided article, explaining each step with clarity and detail:
Combinations Formula:
The combination formula is expressed as: [{}^nC_r = \dfrac{n!}{r!(n - r)!}]
This formula calculates the number of combinations when 'r' elements are selected from a total of 'n' elements. It can also be represented as: [{}^nCr = {}^nC{n - r}]
Problem Description:
Brenda has the option to choose between 2 pairs of pants and 3 shirts. The goal is to determine the number of outfits she can create by selecting one shirt and one pair of pants.
Step-by-Step Solution:
-
Choosing a Shirt:
- Given that Brenda has 3 shirts ((n = 3)) and she needs to choose 1 shirt ((r = 1)).
- Apply the combination formula: [ {}^3C_1 = \dfrac{3!}{1!(3 - 1)!} ]
- Simplify using factorial: [ {}^3C_1 = \dfrac{3 \times 2 \times 1}{1 \times (2 \times 1)} = 3 ]
- Therefore, Brenda can choose a shirt in 3 ways.
-
Choosing a Pant:
- Given that Brenda has 2 pairs of pants ((n = 2)) and she needs to choose 1 pair ((r = 1)).
- Apply the combination formula: [ {}^2C_1 = \dfrac{2!}{1!(2 - 1)!} ]
- Simplify using factorial: [ {}^2C_1 = \dfrac{2 \times 1}{1 \times 1} = 2 ]
- Therefore, Brenda can choose a pair of pants in 2 ways.
-
Total Number of Outfits:
- Multiply the number of ways to choose a shirt by the number of ways to choose a pair of pants: [ \text{Total outfits} = 3 \times 2 = 6 ]
Final Answer:
[ \therefore \text{The number of ways Brenda can choose outfits is 6.} ]
Note on Combinations:
Combination involves selecting a given number of elements taken one by one, or some, or all at a time. Understanding the definition and formula for combinations is crucial, especially when dealing with scenarios where arrangements are not considered. Students should pay attention to the conditions given in the question to avoid confusion in finding arrangements.
