Question

1. In how many distinguishable ways can we rearrange the letters in the word "conversationalists"?

2. A person is asked to draw one card from a standard deck of (well-shuffled) cards, look at the card, place it back into the deck, and then reshuffle the deck. If the experiment is repeated 5 times, find the probability of drawing the Ace of Clubs exactly twice (out of the 5 draws).

Answer 1

1) The number of permutations (ways of arranging) of n things of which p are of one kind, q are alike of another kind, r are alike of another kind = where n = p + q + r

"conversationalists" has 18 letters

letters repeated once - c,v,e,r,l -- 5 letters

letters repeated twice - o,i,n,a,t -- 5 letters

letters repeated thrice - s -- 1 letter

The number of distinguishable ways we can rearrange the letters in the word "conversationalists" =

2) There is one ace of club card in a standard deck of 52 cards

Therefore the probability of drawing an ace of club card = 1/52

Let X be the number of times the draw resulted in the ace of clubs out of the 5 draws

X follows binomial distribution with parameters p = 1/52 and n = 5

where x = 0,1,2,...n

where x = 0,1,2,3,4,5

Therefore the probability of drawing the ace of clubs exactly twice = 0.0035