Pairs All

Pairs All

What is the probability that all pairs will be made up of one red and one blue sock?

A drawer contains n and red socks blue. Socks are taken randomly paired without replacement. What is the probability that all pairs will be made up of one red and one blue sock?

Here's a short answer, followed by my original work on the problem: # 1 Tirera socks, socks # 2, # 3 socks, sock # 4 … sock # (2n-1), sock # (2n) Total number of orders is (2n)! For each pair (1-2) (3-4) … (2n-1), (2n), one of them must be red. Thus, for each pair, there are two choices for the red sock. Then, the rest are blue. Now we have assigned each sock position order a color in one of 2 ^ n ways. Then we can assign individual socks, first the Reds in n! ways, then No blues! ways. While the total number of inconsistent arrangements-even-is 2 ^ nn! No! and the likelihood of this happening at random is 2 ^ n! No! / (2n)! ——————————————— ————— And for the benefit of those who would like to know that solutions concise not just pop out of the air, here's how I approached the problem. —————– ——————————— ———- First sock is a little color. Second round should be another color: (N) / (2n-1) Third sock is some color: Fourth sock color must be different: (n-1) / (2n-3) and so on but a sock last one color. Last socks must be another color: 1 / 1 multiplied together: (n) (n-1) … (1) / (2n-1) (2n-3) … (3) (1) Numerator is n! The denominator is (2n-1)! / (2n -2) (2n -4) (2n-6) … 2 Sub-denominator is 2 (n-1) 2 (n-2) 2 (n-3) … 2 = 2 ^ (n-1) (n-1)! Final Forms ——- ——- We have therefore 2 ^ (n-1) n! (n-1)! / (2n -1)! Multiplying by 2n/2n gives 2 ^ (n) (n!) ^ 2 / (2n)! ————— ——————— The 2 ^ n is the that you can get either red or blue first, if every incompatible pair can come in 2 orders. Ivan t ignore it. His answer has the number of arrangements where the sock is always the same color. Example: 3 pairs given formula: 2 ^ 2 3! 2! / 5! = 4 * 6 * 2 / 120 = 48/120 = 2 / 5 or 2 ^ 3 3! 3! / 6! = 288 / 720 = 2 / 5 Take a sock, say that it is red. Next should be blue: 3 / 5 Take a sock, say it blue. Next should be red: 2 / 3 Now just 1 red, 1 blue left 3. 5 * 2 / 3 = 2 / 5 arrangements Total: 6! = 720 sandwich arrangements: 6 * 3 * 4 * 2 * 2 * 1 = 288 288 / 720 = 2 / 5

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