You are packing for a long-overdue vacation. After all the essential items have
ID: 3041324 • Letter: Y
Question
You are packing for a long-overdue vacation. After all the essential items have been packed, there are still twelve clothing items on your bed that you could take on the trip (four pairs of pants, five shirts, and three pairs of socks). However, there is only room for another four pieces of clothing in your suitcase (a) How many different selections of four clothing items could be packed? b) How many selections result in exactly two pairs of pants being packed? c) How many selections result in at least one item of each type being packed? (d) You grab four items at random, stuff them in your suitcase, and head off to the airport for your flight. What is the probability that you didn't add any socks to your luggage when randomly selecting tems for inclusion!Explanation / Answer
Please note nCx = n! / [(n-x)!*x!]
There are 4 pants + 5 shirts + 3 pairs socks = 12 items
(a) Different selections of 4 items = 12C4 = 12!/(8!x4!) = 495 ways
(b) Exactly 2 pairs of pants: we can choose 2 pants in 4C2 = 6 ways. From the 5 shirts and 3 socks which are 8 items, we can choose 2 in 8C2 = 28 ways.
Therefore total number of ways = 6 x 28 = 168 ways
(c) We can select 1 pant in 4C1 = 4 ways, 1 shirt in 5C1 = 5 ways and 1 pair of socks in 3C1 ways. Out of the remaining 9 (3 pants, 4shirts, 2 socks), we can select 1 in 9C1 = 9 ways. Therefore total number of ways = 4 x 5 x 3 x 9 = 540 ways
(d) Probability = Number of favourable outcomes/Total number of outcomes.
Here total is as calculated in (a) = 495 ways
We now find the number of ways of selecting at least 1 pair of socks = (1 socks from 3 and 3 items from the remainig 9) + (2 socks from 3 and 2 items from the remainig 9) + (3 socks from 3 and 1 items from the remainig 9) = (3C1 x 9C3) + (3C2 x 9C2) + (3C3 x 9C1) = (3 x 84) + (3 x 36) + (1 x 9) = 252 + 108 + 9 = 369 ways.
Therefore the number of selections in which we will have no socks = 495 - 369 = 126 ways.
Therefore the required probability = 126/495 = 0.2545
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