Solution - Combinations without repetition

90

See steps

Step-by-step explanation

1. Find the number of terms in the set

$n$ represents the total number of items in the set:

$c (n, k)$

$c (90, 1)$

$n = 90$

2. Find the number of items selected from the set

$k$ represents the number of items selected from the set:

$c (n, k)$

$c (90, 1)$

$k = 1$

3. Calculate the combinations using the formula

Plug $n$ ( $n$ =90) and $k$ ( $k$ =1) into the combination formula:
$C (n, k) = \frac{n!}{k! (n - k)!}$

6 additional steps

$C (90, 1) = \frac{90!}{1! (90 - 1)!}$

$C (90, 1) = \frac{90!}{1! \cdot 89!}$

$C (90, 1) = \frac{90 \cdot 89 \cdot 88 \cdot 87 \cdot 86 \cdot 85 . . . 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1!}{1! \cdot 89!}$

$C (90, 1) = \frac{90 \cdot 89 \cdot 88 \cdot 87 \cdot 86 \cdot 85 . . . 5 \cdot 4 \cdot 3 \cdot 2}{89!}$

$C (90, 1) = \frac{90 \cdot 89 \cdot 88 \cdot 87 \cdot 86 \cdot 85 . . . 5 \cdot 4 \cdot 3 \cdot 2}{89 \cdot 88 \cdot 87 \cdot 86 \cdot 85 \cdot 84 . . . 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

$C (90, 1) = \frac{90 * 89 * 85 . . . 5 * 2}{89 * 85 * 84 . . . 5 * 2 * 1}$

$C (90, 1) = 90$

There are 90 ways that 1 items chosen from a set of 90 can be combined.

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Why learn this

Combinations and permutations

If you have 2 types of crust, 4 types of toppings, and 3 types of cheese, how many different pizza combinations can you make?
If there are 8 swimmers in a race, how many different sets of 1st, 2nd, and 3rd place winners could there be?
What are your chances of winning the lottery?

All of these questions can be answered using two of the most fundamental concepts in probability: combinations and permutations. Though these concepts are very similar, probability theory holds that they have some important differences. Both combinations and permutations are used to calculate the number of possible combinations of things. The most important difference between the two, however, is that combinations deal with arrangements in which the order of the items being arranged does not matter—such as combinations of pizza toppings—while permutations deal with arrangements in which the order the items being arranged does matter—such as setting the combination to a combination lock, which should really be called a permutation lock because the order of the input matters.

What these two concepts have in common, is that they both help us understand the relationships between sets and the items or subsets that make up those sets. As the examples above illustrate, this can be used to better understand many different types of situations.

Terms and topics

Combinations and Permutations