Ryan reaches into his gym bag for a pair of running shoes. If there are seven different pairs of shoes loose in the bag, how many ways can he pick two shoes that do not match?
Number of ways he can choose two shoes
=14C2
=14!/(2!12!)
=14*13/2
=91
Number of ways he can choose matching shoes
= 7
So how many ways can he choose non-matching shoes?
adawdawd
To find the number of ways Ryan can pick two shoes that do not match from the seven different pairs of shoes, we need to account for two scenarios:
Scenario 1: Ryan picks two shoes of the same pair.
In this scenario, Ryan needs to select one pair of shoes out of the seven pairs available. He can do this in 7 different ways.
Scenario 2: Ryan picks two shoes from different pairs.
In this scenario, Ryan needs to select one pair of shoes out of the seven available, which is the first shoe he picks. Then, for the second shoe, he needs to select one pair out of the remaining six pairs available. Ryan can do this in 7 x 6 = 42 different ways.
Now, to find the total number of ways Ryan can pick two shoes that do not match, we can add the number of ways from each scenario together:
7 (scenario 1) + 42 (scenario 2) = 49.
Therefore, Ryan can pick two shoes that do not match in 49 different ways.
To find the number of ways Ryan can pick two shoes that do not match, we can divide the problem into two cases:
Case 1: Two different pairs of shoes
To choose two different pairs of shoes, Ryan needs to select one pair of shoes out of the seven available pairs. After selecting one pair, he needs to pick one shoe from that pair (2 shoes in total). Since there are 7 different pairs, we can use the combination formula to calculate the number of ways to choose a pair:
C(7, 1) = 7
For each pair that Ryan selects, he has 2 choices for the shoes. Therefore, the total number of ways to select two shoes from different pairs is:
7 pairs × 2 choices = 14
Case 2: One pair and two odd shoes
To choose one pair of shoes and two odd shoes, Ryan needs to select one pair of shoes out of the seven available pairs. After selecting one pair, he needs to pick two different shoes from different pairs (3 shoes in total). Using the combination formula, we can calculate the number of ways to choose a pair:
C(7, 1) = 7
For each pair that Ryan selects, there are 6 remaining pairs to choose from for the odd shoes. Once a pair is chosen, there are 2 choices for each odd shoe. Therefore, the total number of ways to select one pair and two odd shoes is:
7 pairs × 6 remaining pairs × 2 choices × 2 choices = 168
Finally, to find the total number of ways Ryan can pick two shoes that do not match, we sum the results from the two cases:
14 + 168 = 182
Hence, Ryan can pick two shoes that do not match in 182 different ways.