Math CentralQuandaries & Queries


Question from John:

How many combinations of 5 numbers be created with 9 numbers?
Sample: 1,4,7,9,12. Question the same with combination of 4 from 12?

Hi John,

You didn't say if order is important. For example is 1,4,7,912 the same as 1,3,9,7,12? I am going to assume that order is important so that the two examples I listed are different. If I am incorrect let me know.

You also didn't say if numbers can be repeated, for example is 1,4,7,4,9 legitimate? lets do it both ways.

Numbers can be repeated:
You have 9 numbers and you want to create combination of 5 of them. You have 9 choices for which number you want to list first. Since numbers can be repeated, regardless of which number you chose first you have 9 numbers to choose from for the second number. Hence you gave $9 \times 9 = 9^2$ choices for the first two numbers. In a similar fashion you now have 9 choices for the third number and hence there are $9 \times 9 \times 9 = 9^3$ choices for three numbers. Hence you can see that there are $9^5$ possibilities with5 numbers.

Numbers can not be repeated:
Again there are 9 choices for the first number you choose. This time since you can't repeat you only have 8 choices for the second number and hence there are $9 \times 8$ possibilities for a 2 number combination. For the third number you only have 7 possibilities and hence $9 \times 8 \times 7$ combinations using 3 numbers. continue up to 5 numbers.

If you want to see the results you can use the Combinations and Permutations Calculator.


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