**Middle School Number Sense Lesson 47: Sets and Subsets**

There are at least 7 different types of concepts concerning

**sets**on number sense tests. Today we'll discuss how to determine the number of

**subsets**a set has. This concept appeared

**17 times**last year, usually at

**question # 47**.

**Number Dojo Level: 193**

A

**set**is a group of numbers, letters, or symbols, and has any number of elements, which can be identified:

- by name: like {h,e,x,a,g,o,n} or {a,b,c,d,e}, or
- by number: like "a 4-element set" or "a set has 8 elements."

A

**subset**is any (usually smaller) group of the set's elements. So some of the subsets of {h,e,x,a,g,o,n} would be {h,e,x}, {a,g,o,n}, {h,g}, {o}, etc.

*In sets and subsets,*

**order does not matter**, so {h,e,x} is the same as {e,h,x} and {x,e,h}, etc.To determine how many total subsets a set has, simply count the number of elements and take the number

**2 to that power**.

**2**subsets,

^{3}(8)a 4-element set has

**2**subsets,

^{4}(16)a 5-element set has

**2**subsets, etc.

^{5}(32)**"How many 2-element subsets does {f,i,v,e} have?**To calculate this, you can count them:

- {f, i}
- {f, v}
- {f, e}
- {i, v}
- {i, e}
- {v, e}

**combinations**, which we will cover in more detail in a future lesson. I will give a quick overview here--just know that you will need to have a good understanding of

**FactorialOper**.

If a set has

**n**(number of) elements, and you want to know how many subsets of

**r**(number of) elements it has, take

**n!/(r!)(n - r)!**

So in the above example of a

**4**-element set and

**2**-element subsets, take

**4!/(2!)(4 - 2)!**. This simplifies to 4!/(2!)(2!), which is 24/(2 x 2), which is

**6**. Notice that's how many we counted above.

**Example 1: The set {a,b,c,d,e} has how many subsets?**

- Count the number of elements:
**5** - 2 to the 5th power is
**32**. This is your answer.

**Example 2: The set {h,e,x,a,g,o,n} has ___ subsets**

- Count the number of elements:
**7** - 2 to the 7th power is
**128**.

**Example 3: An 8-element set has ___ subsets**

- 2 to the 8th power is
**256**.

**Example 4: The set {l,e,m,o,n,s} has how many 3-element subsets?**

- n = 6, and r = 3. Think of the formula:
**n!/r!(n-r)!** - 6!/3!(6 - 3)! simplifies to
**6!/3!3!**6! is 720, and 3! is 6. - 720/(6 x 6) = 720/36 =
**20**.

**Example 5: A set with 8 elements has how many 3-element subsets?**

- n = 8 and r = 3.
- 8!/3!(8 - 3)! = 8!/3!5!
- 8! is the same as 8 x 7 x 6 x 5!. Cancel out the 5! from the top & bottom. So (8 x 7 x 6 x 5!)/3!5! is the same as (8 x 7 x 6)/6.
- Cancel out the 6's. The answer is 8 x 7 =
**56**.

**Example 6: A 5-element set has ___ subsets containing only 2 or 3 subsets**

- Let's look at the 3-element subsets first. n = 5 and r = 3.
- 5!/3!(5 - 3)! =
**5!/3!2!**.*This is the same as***5!/2!3!**, so there are an equal number of 2-element and 3-element subsets in a set with 5 elements. - 5!/3!2! = 120/(6 x 2) = 120/12 =
**10** (3-element subsets). Multiply this by 2 (for the 2-element subsets) to get**20**.

**Here's a free worksheet to help you practice SetsSubsets:**

setssubsets.pdf |

**Up Next for Middle School: AngleExterior**