   SEARCH HOME Math Central Quandaries & 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? Thanks. 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.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.