A camp counselor and six campers are to be seated along a picnic bench. In how many ways can this be done if the counselor must be seated in the sixth seat and a camper who has a tendency to engage in food fights must sit to the counselor's immediate right?
Answer
Final Answer: The number of ways the camp counselor and six campers can be seated along the picnic bench, with the counselor in the sixth seat and the camper who tends to engage in food fights to the counselor's immediate right, is \(\boxed{120}\).
Step 1 ：Let's denote the number of remaining campers as \( n = 5 \) and the number of remaining seats as \( r = 5 \).
Step 2 ：We need to calculate the permutations, which is the number of ways the remaining campers can be seated in the remaining seats.
Step 3 ：The formula for permutations is \( P(n, r) = \frac{n!}{(n-r)!} \).
Step 4 ：Substituting the values of \( n \) and \( r \) into the formula, we get \( P(5, 5) = \frac{5!}{(5-5)!} \).
Step 5 ：Calculating the factorial of 5, we get \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Step 6 ：Since \( (5-5)! = 0! = 1 \), the denominator of the fraction becomes 1.
Step 7 ：So, the number of permutations is \( P(5, 5) = \frac{120}{1} = 120 \).
Step 8 ：Final Answer: The number of ways the camp counselor and six campers can be seated along the picnic bench, with the counselor in the sixth seat and the camper who tends to engage in food fights to the counselor's immediate right, is \(\boxed{120}\).
