Calculate each Factorial:

Question

a) 30x4! The answer is 720 when you plug it in the calculator, but how can make it to be 6!, 6! = 720, if that makes sense.

b) 7 / 2! 5! + 7! / 4! 3!

I did, 7x6x4x3 = 504, 7x6x5 / 3x2x1 = 84, 84 + 504 = 588 but the answer was supposed to be 56.

c) 9! / 3! 6!

I did 9x8x7x5x4 = 10,080 but the answer was 84.

Answer 1

The key here is to remember what factorials look like when "expanded" out. In general,

n! = n x (n-1) x (n-2) x , , , x 2 x 1

or

7! = 7x6x5x4x3x2x1

So, for part a, we can make the expression look like 6! by doing the following

30x4! = 30x4x3x2x1

Notice that 30 = 6x5

30x4x3x2x1 = 6x5x4x3x2x1 = 6!

For part b, if we expand out each factorial, we will see a lot of cancelling between the numerator and denominator. I will give an example of the second half of part b and then let you take it from there.

7! / 4! 3! = 7x6x5x4x3x2x1 / (4x3x2x1)x(3x2x1)

Notice that 3x2x1 = 6

7x6x5x4x3x2x1 / (4x3x2x1)x6

Now notice that there is a 4,3,2,1, and 6 on top and bottom of the fraction, so we can cancel those out,

7x5 = 35

(remember that this is only the second half of part b, you can confirm that the first half is equal to 21, meaning the answer is 56 as you said)

Answer 2

a) 30 x 4! = 6 x 5 x 4! = 6! (30=6x5)

6! = 30 x 4! = 720

either = 6(5)(4)(3)(2)(1) = 30x24= 720

b) 7!/2!5! + 7!/3!4! = 7(6)/2 + 7(6)(5)/3(2) = 21 +35 =56

You seem to have omitted ! after the first 7

c) 9!/3!6! = 9(8)(7)/3(2) = 3(4)7) = 84

9!= 9(8)(7)(6!) the 6! cancels with the 6! in the denominator leaving

9(8)(7)/3(2)(1). 3 divides into 9 = 3 and 2 divides into 8 = 4 leaving

3(4)(7) = 12(7) = 84