A pitcher knows how to throw five different pitches. He threw four pitches before the last batter was called out.

a) assuming he threw each pitch at LEAST once, determine the number of possible arrangements of his pitches.

b) assuming that he can use any pitch an unlimited number of times, determine the number the number of possible arrangements of his pitches.
Stuck on this one :-(

b)

He can throw 5 types of pitches each pitch. With 5 pitches, there are 5^5 possible combinations.

m<1=

To solve this problem, we can use the concept of permutations.

a) Assuming the pitcher threw each pitch at least once, we need to find the number of permutations of 4 items taken from a set of 5. Since we are assuming the pitcher threw each pitch at least once, we can treat it as arranging 4 items in a specific order.

To calculate this, we use the formula for permutations: P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items we want to arrange.

In this case, we have 5 pitches and we need to arrange 4 of them. So the number of possible arrangements is P(5, 4) = 5! / (5-4)! = 5! / 1! = 5*4*3*2 = 120.

Therefore, there are 120 possible arrangements of his pitches if he threw each pitch at least once.

b) Assuming that the pitcher can use any pitch an unlimited number of times, we need to find the number of permutations of 4 items taken from a set of 5, where repetition is allowed.

To calculate this, we use the formula for permutations with repetition: n^r, where n is the total number of items and r is the number of items we want to arrange.

In this case, we have 5 pitches and we need to arrange 4 of them. So the number of possible arrangements is 5^4 = 5*5*5*5 = 625.

Therefore, there are 625 possible arrangements of his pitches if he can use any pitch an unlimited number of times.

Hope this helps!