Problema Solution
In a certain country, television stations names contain either three or five letters and must start with either the letters W, X or Y. How many station names are available if the names do not contain any repeated letters?
Answer provided by our tutors
Since the English alphabet has 26 letters we need to find all the triplets where the first letters are W, X or Y and the rest (2, 3 or 4 letters) are variations of the left 26-1 = 25 letters
For 3-letter stations: There are 3 positions, the first has 3 possibilities, the other three have 25 possibilities, each decreasing by one as to not repeat the previous letters: 3 * 25 * 24 = 1800
For 4-letter stations: There are 4 positions, the first has 3 possibilities, the other three have 25 possibilities, each decreasing by one as to not repeat the previous letters: 3 * 25 * 24 * 23 = 41400
For 5-Letter stations: 3 * 25 * 24 * 23 * 22 = 910800
Total possibilities = 1800 + 41400 + 910800 = 954,000.
Another way to solve this problem is by using variations without repetition:
3* (V(25,2) + V(25,3) + V(25,4)) where
V(n,k) is k-variations without repeating from n elements
V(n,k) = n! / (n-k)!
where n! = 1*2*...*(n-1) by definition
3* (V(25,2) + V(25,3) + V(25,4)) =
= 3 * ( 25!/23! + 25!/22! + 25!/21!) =
= 3* (24*25 + 23*24*25 + 22*23*24*25)=
= 3 * (600 + 13800 + 303600) =
= 954,000