Sock Drawer had 10 white socks and 4 black socks. List and count possible combinations:
1) of two socks, if order matters
2) of two socks, if order doesn't matter
3) of three socks, if order matters
4) of three socks, if order doesn't matter
if order matters, it's a permutation.
Check your text for perms/combs with duplicates.
To find the possible combinations, we can use the concept of permutations and combinations.
1) Combinations of two socks with order matters:
When order matters, we use permutations. The formula to calculate permutations is nPr = n! / (n - r)!, where n is the total number of objects and r is the number of objects we are selecting at a time.
In this case, we have 14 socks (10 white and 4 black). So, the number of combinations of two socks with order matters is:
14P2 = 14! / (14 - 2)! = 14! / 12! = 14 * 13 = 182
2) Combinations of two socks with order doesn't matter:
When order doesn't matter, we use combinations. The formula to calculate combinations is nCr = n! / (r! * (n - r)!)
In this case, the number of combinations of two socks with order doesn't matter is:
14C2 = 14! / (2! * (14 - 2)!) = 14! / (2! * 12!) = 14 * 13 / 2 = 91
3) Combinations of three socks with order matters:
Using the same formula as in case 1, the number of combinations of three socks with order matters is:
14P3 = 14! / (14 - 3)! = 14! / 11! = 14 * 13 * 12 = 2,184
4) Combinations of three socks with order doesn't matter:
Using the same formula as in case 2, the number of combinations of three socks with order doesn't matter is:
14C3 = 14! / (3! * (14 - 3)!) = 14! / (3! * 11!) = 14 * 13 * 12 / 3 * 2 = 364
To summarize:
1) Combinations of two socks with order matters: 182
2) Combinations of two socks with order doesn't matter: 91
3) Combinations of three socks with order matters: 2,184
4) Combinations of three socks with order doesn't matter: 364