 What does an exclamation mark represent in Math?

# Calculations - Factorials - Introduction

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Ah yes, here we are being hit with yet another new math vocabulary word - factorials. You’ll probably be very happy to learn that factorials are really very easy and even fun. Later on as you progress in your math classes such as in trigonometry or calculus, or even physics, you will learn how to use factorials in several different formulas. However, for this quiz, we are just going to get comfortable with understanding the very basics of factorials.

Okay, so what is a factorial? A factorial is the sum of all whole numbers multiplied in ascending order that precede the factorial number until the factorial number is reached. So what does that mean? Well, let’s look at a factorial number. The factorial number is 6!

Did you see what was highlighted? 6! The exclamation point is part of the number. So if you see the number “6” standing alone it is just the number 6 but if you see it written as “6!” then you have the factorial 6!

Now, we learned that the factorial is the sum of all the whole numbers multiplied in ascending order that precede the factorial number. What this now means is that you find all of the whole numbers that precede (come before) the factorial number 6! These include the numbers, 1, 2, 3, 4 and 5. Next we are told that we include the numbers until the factorial number is reached. This means we have 1, 2, 3, 4, 5 and 6.

Next we are told that the factorial is the sum of the whole numbers multiplied in ascending order. This means we need to multiply each number as we ascend to the factorial number so it will look like the following:

1 x 2 x 3 x 4 x 5 x 6

So we then work to find the sum.

1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720 (We stop here as we have reached the factorial number.)

720 is the sum of factorial 6! This math problem would then be written out in the following manner.

6! = 720

It can also be written as the factorial of 6! is 720.

See, it really is pretty easy isn’t it? Well then, if you think you feel ready, then see if you can find what the factorials are equal to in the next ten problems.

1.
5! = ?
120
125
130
135
Working the problem we have
1 x 2 x 3 x 4 x 5
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
Solution: 5! = 120
2.
11! ÷ 5! = ?
332,640
332,620
332,600
332,580
Working the problem we have
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
5,040 x 8 = 40,320
40,320 x 9 = 362,880
362,880 x 10 = 3,628,800
3,628,800 x 11 = 39,916,800
- and then -
1 x 2 x 3 x 4 x 5
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
Solution: 11! ÷ 5! = 39,916,800 ÷ 120 = 332,640
11! ÷ 5! = 332,640
3.
7! - 6! = ?
3,420
4,320
2,430
4,540
Working the problem we have
1 x 2 x 3 x 4 x 5 x 6 x 7
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
- and then -
1 x 2 x 3 x 4 x 5 x 6
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
Solution: 7! - 6! = 5,040 - 720 = 4,320
7! - 6! = 4,320
4.
10! = ?
3,428,800
342,880
3,628,800
362,880
Working the problem we have
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
5,040 x 8 = 40,320
40,320 x 9 = 362,880
362,880 x 10 = 3,628,800
Solution: 10! = 3,628,800
5.
3! = ?
8
4
12
6
Working the problem we have
1 x 2 x 3
1 x 2 = 2
2 x 3 = 6
Solution: 3! = 6
6.
7! = ?
4,050
4,250
5,040
5,080
Working the problem we have
1 x 2 x 3 x 4 x 5 x 6 x 7
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
Solution: 7! = 5,040
7.
4! + 9! = ?
5,660,910
362,904
362,928
566,091
Working the problem we have
1 x 2 x 3 x 4
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
- and then -
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
5,040 x 8 = 40,320
40,320 x 9 = 362,880
Solution: 4! + 9! = 24 + 362,880 = 362,904
4! + 9! = 362,904
8.
6! x 2! = ?
1,442
722
732
1,440
Working the problem we have
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
- and then -
1 x 2
1 x 2 = 2
Solution: 6! x 2! = 720 x 2 = 1,440
6! x 2! = 1,440
9.
8! = ?
35,320
40,320
40,040
404,020
Working the problem we have
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
5,040 x 8 = 40,320
Solution: 8! = 40,320
10.
8! ÷ 4! x 2! = ?
3,160
3,260
3,360
3,460
Working the problem we have
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5,040
5,040 x 8 = 40,320
- and then -
1 x 2 x 3 x 4
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
- and then
1 x 2 = 2
Solution: 8! ÷ 4! x 2!= 40,320 ÷ 24 = 1,680 x 2 = 3,360
8! ÷ 4! x 2!= 3,360