Answer:
1. B ie 26! / 21!
The first letter can be any of 26
The second letter can be any of 25 (not the first)
The third letter can be any of 24 (not the first or second)
The 4th letter can be any of 23 (not the first three)
The 5th letter can be any of 22 (not the first 4)
Therefore there are 26 * 25 * 24 * 23 * 22 permutations.
What is another way of expressing this using factorials?
Well 26! = 26 * 25 * 24 * 23 * 22 * 21 * 20........* 3 * 2 * 1
But we only need the first 5 terms we don't need 21* 20 * 19 * 18 * 17........*3 * 2 * 1
These numbers that we don't need can be expressed as 21!
Therefore if you divide 26! by 21! you get 26 * 25 * 24 * 23 * 22
2. = Answer B
The pitcher bats last = 1 ie no choice
Therefore first can be any of the team except the pitcher = 8 possible names
Second can be any except the first and the pitcher = 7
And so on
8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 * 1 = 8! = 40320
(The final "1" represents the pitcher, where there is no choice.)
3. Answer = B
If you stood people in a line there would be 7! permutations = 7 * 6 * 5 * 4 * 3 * 2 * 1
The first can be any of 7
The second can be any of 6
etc etc
7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5040
BUT you are not putting them in line. You are seating them at a round table. In those 7! permutations there will be 7 identical permutations where all you do is go round the table starting at each of the 7 seats in turn
EG the order ABCDEFG starting at seat 1 going clockwise is an identical order as ABCDEFG starting at seats 2,3,4,5,6 & 7.
Therefore we need to divide the 7! permutations by 7 to reflect this
7! / 7 = 6!
= 6 * 5 * 4 * 3 * 2 * 1 = 720
OR
If there are 5 letters used only, then 5! or 5*4*3*2 = 120
If all 26 letters of the alphabet may be used for the code, then:
26*25*24*23*22
If you mean letters of the alphabet, and not case sensitive, it's26*25*24*23*22
= 7893600
