Final answer:
To calculate the number of distinguishable permutations that can be made with the letters T,O,P,P,S, we divide the factorial of the total number of letters (5!) by the factorial of the number of times each distinct letter appears (for P, it's 2!). This gives us a total of 60 unique permutations.
Step-by-step explanation:
The student has asked, "How many distinguishable permutations can be made using the letters T,O,P,P,S?" To solve this, we use the formula for permutations of a multiset: n! / (n1! * n2! * ... * nk!), where n! is the factorial of the number of items, and n1!, n2!, ..., nk! are the factorials of the number of times each distinct item appears.
In the case of the letters T, O, P, P, S, we have 5 letters in total and the letter 'P' appears twice. Thus, the number of distinguishable permutations would be: 5! / 2!, which is 120 / 2, giving us 60 distinguishable permutations.
Breaking it down further,
5! represents the total number of permutations if all the letters were unique, which is 5*4*3*2*1, or 120. Since the letter 'P' repeats, we divide by the factorial of the count of P, which is 2, to adjust for the repetitions. So we calculate 120 / (2*1), simplifying to 120 / 2 = 60 unique permutations.