30 red socks, 22 gold socks and 14 black socks. How many socks would he have to pick up before getting a matching pair?
There are only 3 colors.
So, 4 socks must have two matching.
10
To determine how many socks one would have to pick up before getting a matching pair, we need to apply the pigeonhole principle. The pigeonhole principle states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon.
In this case, since we are looking for a matching pair of socks, each color can be considered as a pigeonhole. We have 30 red socks, 22 gold socks, and 14 black socks, which gives us a total of 66 socks, spread across different colors.
To find out the minimum number of socks needed to guarantee a matching pair, we need to consider the worst-case scenario. In this case, the worst-case scenario would be if we pick socks of different colors one after another until we have at least one pair. So, let's calculate the maximum number of socks we can pick without getting a matching pair for each color:
- For red socks: Since we have 30 red socks, the maximum number of red socks we can pick without getting a matching pair is 29, as the 30th sock, when picked, will complete a pair.
- For gold socks: Similarly, with 22 gold socks, we can pick at most 21 gold socks without getting a matching pair.
- For black socks: With 14 black socks, we can pick at most 13 black socks without getting a matching pair.
Now, to ensure we have a matching pair, we need to add 1 more sock to the maximum number of socks we can pick without getting a matching pair for each color. Thus, we sum up these values:
29 (maximum red socks) + 21 (maximum gold socks) + 13 (maximum black socks) + 1 (extra sock) = 64 socks
Therefore, one would have to pick up at least 64 socks to guarantee having a matching pair.