Difference between revisions of "MA 375 Spring 2009 Walther Homework 6 Number 10" - Rhea

Revision as of 14:41, 24 February 2009

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How do I start this problem? I'm drawing a complete blank..It seems very familiar like we've done something like this in class or something, but I can't figure it out. Please help. -Brandy

Cool thing about f):

The question calls for the probability that the letter z precedes both a and b in the permutation of the 26 letters of the alphabet. This is basically saying what is the probability that when you mix up all the letters of the alpahabet (a,b,c,..z) that z will come before a and b.

Now, we can solve this the long way (later), or use your head a little: for any permutation of the alphabet, we can circle the letters a, b, and z. These can be arrange in 6 different ways if we just move these three around keeping all others the same. Therefore, the amount of letters we have between a, b, and z don't matter. We can have, 1, 100, or 0. Therefore the question becomes what is the probability that z comes first out of a, b, z. Which the answer is 1/3.

Now, if you don't believe me, here's the long way: we count how many ways that z can be in front of a and b:

if z=1, then a=25 ways and b=24 ways, and 23! others (in which to order them)

if z=2, then a=24 ways and b=23 ways, and 22! others

...

continuing this process to z=24, we get:

$ \left ( 25 \times 24 + 24 \times 23 + 23 \times 22 + ... + 2 \times 1 \right ) \times \frac {23!}{26!} = \frac{5200}{26 \times 25 \times 24} = \frac {1}{3} $