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If 6 times the no. of permutations of n items taken 3 at a time is equal to 7 times the no. of permutations of (n-1) items taken 3 at a time then the value of n will be?
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If 6 times the no. of permutations of n items taken 3 at a time is equ...
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If 6 times the no. of permutations of n items taken 3 at a time is equ...
Given:
6 times the number of permutations of n items taken 3 at a time = 7 times the number of permutations of (n-1) items taken 3 at a time.
To find:
The value of n.
Assumption:
We assume that n is a positive integer.
Solution:
Let's break down the problem into smaller steps:
Step 1:
Write the formula for the number of permutations of n items taken 3 at a time.
The number of permutations is given by the formula: P(n, 3) = n! / (n - 3)!
Step 2:
Apply the given condition to form an equation.
6 * P(n, 3) = 7 * P(n - 1, 3)
Step 3:
Substitute the formula from Step 1 into the equation from Step 2.
6 * (n! / (n - 3)!) = 7 * ((n - 1)! / ((n - 1) - 3)!)
Step 4:
Simplify the equation.
6 * (n! / (n - 3)!) = 7 * ((n - 1)! / (n - 4)!)
Step 5:
Cancel out common factors in the equation.
6 * n! = 7 * (n - 1)!
Step 6:
Expand the factorial terms.
6 * n * (n - 1) * (n - 2) = 7 * (n - 1)!
Step 7:
Cancel out common terms.
6 * n * (n - 2) = 7
Step 8:
Solve the equation for n.
6n^2 - 12n - 7 = 0
Step 9:
Use the quadratic formula to solve for n.
n = (-(-12) ± sqrt((-12)^2 - 4 * 6 * (-7))) / (2 * 6)
n = (12 ± sqrt(144 + 168)) / 12
n = (12 ± sqrt(312)) / 12
Step 10:
Simplify the square root.
n = (12 ± sqrt(4 * 78)) / 12
n = (12 ± 2 * sqrt(78)) / 12
n = 1 ± sqrt(78) / 3
Step 11:
Since n is a positive integer, we discard the negative solution.
n = 1 + sqrt(78) / 3
Final Answer:
The value of n is approximately 3.48.
6 times the number of permutations of n items taken 3 at a time = 7 times the number of permutations of (n-1) items taken 3 at a time.
To find:
The value of n.
Assumption:
We assume that n is a positive integer.
Solution:
Let's break down the problem into smaller steps:
Step 1:
Write the formula for the number of permutations of n items taken 3 at a time.
The number of permutations is given by the formula: P(n, 3) = n! / (n - 3)!
Step 2:
Apply the given condition to form an equation.
6 * P(n, 3) = 7 * P(n - 1, 3)
Step 3:
Substitute the formula from Step 1 into the equation from Step 2.
6 * (n! / (n - 3)!) = 7 * ((n - 1)! / ((n - 1) - 3)!)
Step 4:
Simplify the equation.
6 * (n! / (n - 3)!) = 7 * ((n - 1)! / (n - 4)!)
Step 5:
Cancel out common factors in the equation.
6 * n! = 7 * (n - 1)!
Step 6:
Expand the factorial terms.
6 * n * (n - 1) * (n - 2) = 7 * (n - 1)!
Step 7:
Cancel out common terms.
6 * n * (n - 2) = 7
Step 8:
Solve the equation for n.
6n^2 - 12n - 7 = 0
Step 9:
Use the quadratic formula to solve for n.
n = (-(-12) ± sqrt((-12)^2 - 4 * 6 * (-7))) / (2 * 6)
n = (12 ± sqrt(144 + 168)) / 12
n = (12 ± sqrt(312)) / 12
Step 10:
Simplify the square root.
n = (12 ± sqrt(4 * 78)) / 12
n = (12 ± 2 * sqrt(78)) / 12
n = 1 ± sqrt(78) / 3
Step 11:
Since n is a positive integer, we discard the negative solution.
n = 1 + sqrt(78) / 3
Final Answer:
The value of n is approximately 3.48.
Community Answer
If 6 times the no. of permutations of n items taken 3 at a time is equ...
Options are - a 7 b 13 c 21 d 9
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If 6 times the no. of permutations of n items taken 3 at a time is equal to 7 times the no. of permutations of (n-1) items taken 3 at a time then the value of n will be?
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If 6 times the no. of permutations of n items taken 3 at a time is equal to 7 times the no. of permutations of (n-1) items taken 3 at a time then the value of n will be? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about If 6 times the no. of permutations of n items taken 3 at a time is equal to 7 times the no. of permutations of (n-1) items taken 3 at a time then the value of n will be? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 6 times the no. of permutations of n items taken 3 at a time is equal to 7 times the no. of permutations of (n-1) items taken 3 at a time then the value of n will be?.
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