**High School Number Sense Lesson 56: Solving for X with Factorials**

We have already dedicated a few posts to solving for an unknown in an equation without exponents (

**SolveX**), with exponents (

**SolveXExpo**), and where each term is a fraction (

**SolveXDenom**). Today we will look at solving for X when the equation contains factorials. (You may want to review my

**post**on Factorial Operations before you continue). This concept appeared

**14 times**last year, with a median placement at

**question # 54**.

**Number Dojo Level: TBD**

If you will recall, a factorial of a number is designated with an

**!**after the number. It tells us to multiply that number by each positive integer less than itself. So to solve these equations, we need to keep that in mind (and use a little logic). I think I'll jump straight to some examples:

**Example 1: If 3!(5!) = k!, then k = ___**

- In this example, 3! equals
**6**. - We are multiplying 5! by 6. This is the same as saying
**6!**, so k equals**6**.

**Example 2: If 8! + 4 x 6! = k(6!), then k = ___**

- Here, we will factor out
**6!**from each term in the equation. It becomes: (8 x 7 x 6!) + 4 x 6! = k(6!). - Remove the
**6!**from each term to get**(8 x 7) + 4 = k**. - This simplifies to
**56 + 4 = k**, which becomes**60 = k**. So our answer is**60**.

**Example 3: If 2!3! = 4!/k, then k = ___**

- We should have the first several factorials memorized, so let's use that knowledge.
- Change this to:
**(2)(6) = (24)/k**. - Simplified, this becomes
**12 = 24/k**. k =**2**.

**Example 4: Let 7!/6! = x!/(x - 1)!. Find x. ___**

- 7!/6! =
**7**, and x! =**x(x - 1)!**. Substitute these back into the equation. - We now have
**7 = x(x - 1)!/(x - 1)!**. The (x - 1)! cancels from the top & bottom. - We have
**7 = x**, so our answer is**7**.

**Example 5: Let 6!/4! = x!/(x - 1)!. Find x. ___**

- 6! =
**(6 x 5 x 4!)**. We learned from Example 4 that x!/(x - 1)! is the same as**x**. - We now have
**(6 x 5 x 4!)/4! = x**. The 4! cancels from the top & bottom. - We are left with
**6 x 5 = x**. So**x = 30**.

**Example 6: Let 5!/6! = x!/(x - 1)!. Find x. ___**

- The left side simplifies to
**1/6**. The right side simplifies to**x**. - The answer is
**1/6**.

**Example 7: If 7!/5! = (x + 2)!/(x + 1)!, then x = ___**

- The left side simplifies to
**(7 x 6)**. The right side simplifies to**(x + 2)**. - We now have
**7 x 6 = x + 2**. This becomes**42 = x + 2**. - The answer is
**40**.

**Example 8: Let 9!/8! = x!/(x + 1)!. Find x. ___**

- The left side simplifies to
**9**. The right side simplifies to**1/(x + 1)**. - We have
**9 = 1/(x + 1)**. Multiply both sides by**(x + 1)**. - Now we have
**9(x + 1) = 1**. Simplify this to become**9x + 9 = 1**. - Subtract 9 from both sides to get
**9x = -8**. - Divide both sides by 9 to get
**x = -8/9**.

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

solvexfacto.pdf |

**Up Next for High School: DivMultSquare**