F. Combinations
Suppose
that a job has two different parts. There are m different
ways of doing the first part, and there are n different
ways of doing the second part. The problem is to find the number
of ways of doing the entire job. For each way of doing the first
part of the job, there are n ways of doing the second
part. Since there are m ways of doing the first part,
the total number of ways of doing the entire job is m
x n. The formula that can be used is
Number of ways = m x n
For any problem that involves
two actions or two objects, each with a number of choices, and
asks for the number of combinations, the above formula can be
used.
Example 13
William wants a sandwich and a drink for lunch. If a restaurant
has 4 choices of sandwiches and 3 choices of drinks, how many
different ways can he order his lunch?
Solution
Since there are 4 choices of sandwiches and 3 choices of drinks,
using the formula
Number of ways = 4(3) = 12
Therefore, the man can order
his lunch 12 different ways.
Now, what if the combinations
available decrease after each combination is taken?
Example 14
There are five meal options in the cafeteria of a certain school.
Assuming that a different meal must be eaten each day, and each
different type of meal must be eaten once before any type of
meal can be eaten a second time, how many different orders of
meals can a student eat in the first five days?
Solution
The answer is 120. There are five types of meals, so the total
number of possibilities is 5!. (the "!" stands for
factorial), or 5 x 4 x 3 x 2 x 1 = 120. Since a different meal
is assigned to every day, you must reduce the amount that you
multiply by on a daily basis from 5 to 4 to 3 to 2 to 1. If you
like, assign the letters A, B, C, D and E to the five meals and
see how many different orders you can create.
What if two or more samples are
chosen at a time? If we have objects a, b, c, d and want to arrange
them two at a time--that is, ab, bc, cd,
etc. (We have four combinations taken two at a time). If you
have four different combinations taken two at a time, you can
write this as 4 C 2, which can be written as
4 C 2 = 4 x 3
2
x 1
Examples
8 C 3 = 8 x 7 x 6
3
x 2 x 1
10 C 4 = 10 x 9 x 8 x 7
4
x 3 x 2 x 1
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