How Many Ways Can 5 Paintings Be Lined Up on a Wall

Learning Outcomes

• Detect the number of combinations of n distinct choices.

So far, nosotros have looked at problems request us to put objects in order. There are many bug in which nosotros want to select a few objects from a group of objects, but nosotros practise not intendance about the order. When nosotros are selecting objects and the society does not matter, we are dealing with combinations. A option of [latex]r[/latex] objects from a set of [latex]due north[/latex] objects where the lodge does not matter can be written as [latex]C\left(n,r\right)[/latex]. Just equally with permutations, [latex]\text{C}\left(northward,r\right)[/latex] can besides be written as [latex]{}_{north}{C}_{r}[/latex]. In this case, the general formula is as follows.

[latex]\text{C}\left(n,r\right)=\dfrac{n!}{r!\left(n-r\right)!}[/latex]

An earlier problem considered choosing 3 of iv possible paintings to hang on a wall. Nosotros constitute that there were 24 ways to select 3 of the iv paintings in order. Just what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the aforementioned as selecting paintings 2, 3, i. To discover the number of ways to select 3 of the iv paintings, disregarding the order of the paintings, divide the number of permutations past the number of ways to order 3 paintings. There are [latex]3!=3\cdot 2\cdot 1=6[/latex] ways to order 3 paintings. In that location are [latex]\frac{24}{six}[/latex], or four ways to select three of the iv paintings. This number makes sense because every time we are selecting 3 paintings, nosotros are not selecting ane painting. There are 4 paintings we could choose non to select, and then at that place are four ways to select three of the 4 paintings.

A Full general Note: Formula for Combinations of northward Distinct Objects

Given [latex]northward[/latex] distinct objects, the number of ways to select [latex]r[/latex] objects from the ready is

[latex]\text{C}\left(n,r\right)=\dfrac{due north!}{r!\left(n-r\right)!}[/latex]

How To: Given a number of options, determine the possible number of combinations.

1. Place [latex]northward[/latex] from the given information.
2. Place [latex]r[/latex] from the given information.
3. Supersede [latex]due north[/latex] and [latex]r[/latex] in the formula with the given values.
4. Evaluate.

Case: Finding the Number of Combinations Using the Formula

A fast food restaurant offers 5 side dish options. Your meal comes with ii side dishes.

1. How many ways can you select your side dishes?
2. How many ways tin can you select 3 side dishes?

Q & A

Is it a coincidence that parts (a) and (b) in Example 4 take the same answers?

No. When we cull r objects from n objects, we are not choosing [latex]\left(n-r\right)[/latex] objects. Therefore, [latex]C\left(n,r\right)=C\left(n,n-r\right)[/latex].

Try It

An ice cream store offers x flavors of ice cream. How many ways are at that place to choose iii flavors for a banana separate?

Show Solution

[latex]C\left(10,three\right)=120[/latex]

Finding the Number of Subsets of a Set

We take looked simply at combination problems in which we chose exactly [latex]r[/latex] objects. In some bug, we desire to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can exist ordered. How many different pizzas are possible?

To answer this question, we need to consider pizzas with any number of toppings. There is [latex]C\left(v,0\right)=i[/latex] way to society a pizza with no toppings. In that location are [latex]C\left(5,one\correct)=5[/latex] means to gild a pizza with exactly one topping. If we continue this process, we get

[latex]C\left(five,0\right)+C\left(5,i\correct)+C\left(five,two\right)+C\left(5,3\right)+C\left(5,4\correct)+C\left(5,5\correct)=32[/latex]

In that location are 32 possible pizzas. This result is equal to [latex]{2}^{5}[/latex].

We are presented with a sequence of choices. For each of the [latex]due north[/latex] objects we have two choices: include information technology in the subset or non. So for the whole subset we accept made [latex]n[/latex] choices, each with ii options. So there are a total of [latex]2\cdot 2\cdot 2\cdot \dots \cdot 2[/latex] possible resulting subsets, all the fashion from the empty subset, which we obtain when we say "no" each time, to the original set itself, which nosotros obtain when we say "aye" each time.

A General Annotation: Formula for the Number of Subsets of a Set

A set containing due north distinct objects has [latex]{two}^{north}[/latex] subsets.

Case: Finding the Number of Subsets of a Set

A restaurant offers butter, cheese, chives, and sour foam as toppings for a baked potato. How many different ways are there to club a potato?

Endeavor It

A sundae bar at a wedding has 6 toppings to cull from. Any number of toppings can be chosen. How many different sundaes are possible?

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Source: https://courses.lumenlearning.com/waymakercollegealgebra/chapter/combinations/