30. Parts a, b, c Answer the question posed in (b) for four-character sequences.

Question

30. Parts a, b, c Answer the question posed in (b) for four-character sequences. As of April 2006, 97,786 of the four-character sequences using either letters or digits had not yet been claimed. If a four-character name is randomly selected, what is the probability that it is already owned? A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this? If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? If 6 bottles are randomly selected, what is the probability that all of them are the same variety? The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano).

Explanation / Answer

Solution:

total bottles=30

(a) need 3 bootles of zinfandel

total zinfandel=8

order is imp .go for permuattion

no of ways is 8P3=8!/(8-3)!=8!/5!=336 ways

since order is important.

go for permutation.

(b)order is not important.problem is of combinatiions.

You have 30 bottles and have to select (choose) 6 from them. The number of ways of choosing m from n is nCm =(n!)/[m!(n-m)!] and in this case
30C6=30!/(6!24!)=593775

(c)problem of combinations.
c) (2 from 8)×(2 from 10)×(2 from 12)
=8C2×10C2×12C2

=83160 ways

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