Welcome to combinatorics! Today we'll explore the difference between C(5,3) and A(5,3). C(5,3) represents combinations - choosing 3 items from 5 where order doesn't matter. A(5,3) represents permutations - arranging 3 items from 5 where order does matter. Let's see how to calculate both.
Let's calculate C(5,3) - combinations. The formula is C(n,k) equals n factorial divided by k factorial times n minus k factorial. For C(5,3), we have 5 factorial divided by 3 factorial times 2 factorial. That's 120 divided by 6 times 2, which equals 120 divided by 12, giving us 10. Here are all 10 possible combinations when choosing 3 items from 5, where order doesn't matter.
Now let's calculate A(5,3) - permutations. The formula is A(n,k) equals n factorial divided by n minus k factorial. For A(5,3), we have 5 factorial divided by 2 factorial. That's 120 divided by 2, which equals 60. Here are some example permutations where order matters. Notice that 1,2,3 and 3,2,1 are different permutations, unlike combinations where they would be the same.
The key difference is whether order matters. In combinations C(5,3), we get 10 ways because order doesn't matter - choosing items 1, 2, and 3 is the same regardless of the sequence. In permutations A(5,3), we get 60 ways because order matters - arranging 1,2,3 versus 3,2,1 creates different outcomes. The relationship is A(n,k) equals C(n,k) times k factorial, since each combination can be arranged in k factorial different ways.
To summarize: C(5,3) equals 10 combinations where order doesn't matter, and A(5,3) equals 60 permutations where order does matter. Use combinations when selecting items like choosing team members or picking items from a menu. Use permutations when arrangement matters, like creating passwords, determining race finishing order, or seating arrangements. Remember the key relationship: A(n,k) equals C(n,k) times k factorial.