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Basic Mathematics-II
Notes 12! = 1 × 2 × 3 × 4 ×........ × 12
12! = 479001600
When you begin performing combinations, permutations, and probability, you’ll be evaluating
expressions that have factorials in the numerators and the denominators.
Example: Simplify the following:
6!
4!
From the definition of a factorial:
6! 1 2 3 4 5 6 1 2 3 4 5 6
5 6 30
4! 1 2 3 4 1 2 3 4
Thus 6! ÷ 4! = 30
Simplify the following:
17!
14!3!
At once, you can cancel off the factors 1via 14 that will be common to both 17! and 14!. Then you
can simplify what’s left to obtain:
17! 1 2 3 4 14 15 16 17 1 2 3 4 14 15 16 17 15 16 17
14!3! 1 2 3 4 14 1 2 3 1 2 3 4 14 1 2 3 1 2 3
15 16 17
5 8 17 680
1 2 3
Observe how we reduced what we had to write by leaving a gap (the “ellipsis”, or triple-period)
in the center. This gap-and-cancel process will turn out to be handy later on (such as in calculus,
where you’ll utilize this technique a lot), especially when you’re relating with expressions that
your calculator can’t manage.
Example: Simplify the following:
n 2 !
n 1 !
To do this, we will write out the factorials, by means of enough of the factors to have stuff that
can terminate off. The factors in the product (n + 2)! are of the form:
1 × 2 × 3 × 4 ×...× (n – 1) × (n) × (n + 1) × (n + 2)
Now we have created a list of factors that can cancel out:
n
n 2 ! 1 2 3 n 1 n 1 n 2
n 1 ! 1 2 3 n 1
1 2 3 n 1 n 1 n 2
3
2
n n 1 n 2 n 3n 2n
1 2 3 n 1
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