**High School Number Sense Lesson 43: Factorial Operations**

Factorials show up in different areas of mathematics--most notably when calculating combinations and permutations (which we will cover later). On number sense tests, we are often asked to perform calculations with factorials. This concept appeared

**16 times**this year, with a median placement at

**question # 41**.

**Number Dojo Level: 270**

The factorial (denoted by

**) of an integer**

*n!***(greater than 0) is the product of all positive integers less than or equal to**

*n***. So:**

*n***1!**=

**1**

**2!**= 2 x 1 =

**2**

**3!**= 3 x 2 x 1 =

**6**

**4!**= 4 x 3 x 2 x 1 =

**24**

**5!**= 5 x 4 x 3 x 2 x 1 =

**120**

**6!**= 6 x 5 x 4 x 3 x 2 x 1 =

**720**

**7!**= 7 x 6 x 5 x 4 x 3 x 2 x 1 =

**5040**

**8!**= 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 =

**40320**

and so on.

**0!--which is 1**. Notice that in each case,

**n! = n(n - 1)!**So in other words,

**. This property comes in handy on many questions.**

*every factorial has, as a factor, each factorial less than that number*Also, when performing operations, it is important to take a moment to combine (or cancel out) factors at any opportunity--before performing the operations. I will illustrate what I mean:

**Example 1: 4! + 5!**

- This question can be solved quickly if you have the factorials memorized. 4! =
**24**and 5! =**120**. - 24 + 120 =
**144**.

**Example 2: 11! ÷ 8!**

- When dividing factorials, remember that a larger factorial has, as a factor, every factorial less than that number. So in this case. 11! = 11 x 10 x 9 x 8!.
- Cancel out the 8! from the 11! and the 8!. You are left with 11 x 10 x 9, which is
**990**.

**Example 3: 6! + 5 x 5!**

- Think of this as
**(6 x 5!) + (5 x 5!)**. - Factor out the 5!, which gives you
**(6 + 5)(5!)**. - Simplify this to
**11 x 5!**, which is**11 x 120**. The answer is**1320**.

**Example 4: (5! + 3!) ÷ 4! = ___ mixed number**

- Factor out the
**3!**. You then have**[(5 x 4 x 3!) + 3! ] ÷ (4 x 3!)**. - You are left with
**(5 x 4 + 1) ÷ 4**. This simplifies to (20 + 1)/4 =**21/4**. - Change this to a mixed number. Your answer is
**5 1/4**.

**Example 5: 5!/(3!2!)**

- Factor out the 3! from the top & bottom of the fraction. You then have
**(5 x 4 x 3!)/(3!2!)**. - You are left with
**(5 x 4)/2!**, which is**20/2**. Your answer is**10**.

**Example 6: (7! 4! 3!) ÷ (6! 5! 2!) = ___ decimal**

- Factor the (bottom) 6! from the (top) 7!, the (top) 4! from the (bottom) 5!, and the (bottom) 2! from the (top) 3!.
- You are left with
**[(7 x 6!)(4!)(3 x 2!)] ÷ [(6!)(5 x 4!)(2!)]**. - Remove the common factors
**(6!, 4!, and 2!)**from both the top and bottom of the fraction. - You are left with (7 x 3) ÷ (5), or
**21/5**. - Change this to a decimal to get
**4.2**.

**Example 7: 8!/5! - 7!/4!**

- Factor out the 5! from the first fraction and the 4! from the second fraction.
- You have
**(8 x 7 x 6)(5!)/5! - (7 x 6 x 5)(4!)/4!** - Remove the common factors. You are left with
**(8 x 7 x 6) - (7 x 6 x 5)**. - Notice the new common factors of 7 x 6. Factor them out to get:
**(7 x 6)(8 - 5)**. - Simplify to get
**(42)(3)**, which is**126**.

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

factorialoper.pdf |

**Up Next for High School: TriangleRight**