Answer :
To solve the problem, we need to find [tex]\( n \)[/tex] such that the ratio of the permutations [tex]\( P(n, 4) \)[/tex] to [tex]\( P(n, 3) \)[/tex] equals 9:1. We'll use the formulas for permutations and set up an equation based on the given ratio.
### Step-by-Step Solution
1. Recall the formula for permutations:
[tex]\[ P(n, k) = \frac{n!}{(n-k)!} \][/tex]
where [tex]\( P(n, k) \)[/tex] represents the number of permutations of [tex]\( n \)[/tex] objects taken [tex]\( k \)[/tex] at a time.
2. Write the expressions for [tex]\( P(n, 4) \)[/tex] and [tex]\( P(n, 3) \)[/tex]:
[tex]\[ P(n, 4) = \frac{n!}{(n-4)!} \][/tex]
[tex]\[ P(n, 3) = \frac{n!}{(n-3)!} \][/tex]
3. Form the ratio [tex]\( \frac{P(n, 4)}{P(n, 3)} \)[/tex] and set it equal to 9:
[tex]\[ \frac{P(n, 4)}{P(n, 3)} = 9 \][/tex]
Substitute the expressions of [tex]\( P(n, 4) \)[/tex] and [tex]\( P(n, 3) \)[/tex] into the equation:
[tex]\[ \frac{\frac{n!}{(n-4)!}}{\frac{n!}{(n-3)!}} = 9 \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{n!}{(n-4)!} \times \frac{(n-3)!}{n!} = 9 \][/tex]
[tex]\[ \frac{(n-3)!}{(n-4)!} = 9 \][/tex]
5. Further simplify by canceling out [tex]\( (n-4)! \)[/tex]:
[tex]\[ (n-3) = 9 \][/tex]
6. Solve for [tex]\( n \)[/tex]:
[tex]\[ n - 3 = 9 \][/tex]
[tex]\[ n = 12 \][/tex]
So, the value of [tex]\( n \)[/tex] that satisfies the given ratio is [tex]\( \boxed{12} \)[/tex].
### Verification
To verify, we calculate [tex]\( P(12, 4) \)[/tex] and [tex]\( P(12, 3) \)[/tex] and check if their ratio is indeed 9.
- Calculate [tex]\( P(12, 4) \)[/tex]:
[tex]\[ P(12, 4) = \frac{12!}{(12-4)!} = \frac{12!}{8!} \][/tex]
- Calculate [tex]\( P(12, 3) \)[/tex]:
[tex]\[ P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} \][/tex]
- Form the ratio:
[tex]\[ \frac{P(12, 4)}{P(12, 3)} = \frac{\frac{12!}{8!}}{\frac{12!}{9!}} = \frac{12! \times 9!}{8! \times 12!} = \frac{9!}{8!} = 9 \][/tex]
Thus, the ratio is indeed 9, verifying that our solution [tex]\( n = 12 \)[/tex] is correct.
### Step-by-Step Solution
1. Recall the formula for permutations:
[tex]\[ P(n, k) = \frac{n!}{(n-k)!} \][/tex]
where [tex]\( P(n, k) \)[/tex] represents the number of permutations of [tex]\( n \)[/tex] objects taken [tex]\( k \)[/tex] at a time.
2. Write the expressions for [tex]\( P(n, 4) \)[/tex] and [tex]\( P(n, 3) \)[/tex]:
[tex]\[ P(n, 4) = \frac{n!}{(n-4)!} \][/tex]
[tex]\[ P(n, 3) = \frac{n!}{(n-3)!} \][/tex]
3. Form the ratio [tex]\( \frac{P(n, 4)}{P(n, 3)} \)[/tex] and set it equal to 9:
[tex]\[ \frac{P(n, 4)}{P(n, 3)} = 9 \][/tex]
Substitute the expressions of [tex]\( P(n, 4) \)[/tex] and [tex]\( P(n, 3) \)[/tex] into the equation:
[tex]\[ \frac{\frac{n!}{(n-4)!}}{\frac{n!}{(n-3)!}} = 9 \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{n!}{(n-4)!} \times \frac{(n-3)!}{n!} = 9 \][/tex]
[tex]\[ \frac{(n-3)!}{(n-4)!} = 9 \][/tex]
5. Further simplify by canceling out [tex]\( (n-4)! \)[/tex]:
[tex]\[ (n-3) = 9 \][/tex]
6. Solve for [tex]\( n \)[/tex]:
[tex]\[ n - 3 = 9 \][/tex]
[tex]\[ n = 12 \][/tex]
So, the value of [tex]\( n \)[/tex] that satisfies the given ratio is [tex]\( \boxed{12} \)[/tex].
### Verification
To verify, we calculate [tex]\( P(12, 4) \)[/tex] and [tex]\( P(12, 3) \)[/tex] and check if their ratio is indeed 9.
- Calculate [tex]\( P(12, 4) \)[/tex]:
[tex]\[ P(12, 4) = \frac{12!}{(12-4)!} = \frac{12!}{8!} \][/tex]
- Calculate [tex]\( P(12, 3) \)[/tex]:
[tex]\[ P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} \][/tex]
- Form the ratio:
[tex]\[ \frac{P(12, 4)}{P(12, 3)} = \frac{\frac{12!}{8!}}{\frac{12!}{9!}} = \frac{12! \times 9!}{8! \times 12!} = \frac{9!}{8!} = 9 \][/tex]
Thus, the ratio is indeed 9, verifying that our solution [tex]\( n = 12 \)[/tex] is correct.
