Question 17 of 42intermediate🔧 ApplyNumerical3 marks

Find n if n-1P₃ : nP₄ = 1 : 9.

Correct Answer

n = 9

Exercise: EXERCISE 6.3 | Q: 6 | (Page 15)
For More Understanding

Explanation

This question tests the application of the permutation formula. Using the formula from the textbook context, we express both permutations in factorial form and simplify the ratio equation to find n.

Solution Steps

  1. Step 1: Write the given ratio using the permutation formula. Given: $^{n-1}P_3 : ^nP_4 = 1 : 9

    Using the formula (^nP_r = \frac{n!}{(n-r)!}$, we get:

    n−1P3=(n−1)!(n−1−3)!=(n−1)!(n−4)!^{n-1}P_3 = \frac{(n-1)!}{(n-1-3)!} = \frac{(n-1)!}{(n-4)!} nP4=n!(n−4)!^nP_4 = \frac{n!}{(n-4)!}

  2. Step 2: Set up the ratio equation.

    n−1P3nP4=19\frac{^{n-1}P_3}{^nP_4} = \frac{1}{9}

    (n−1)!(n−4)!×(n−4)!n!=19\frac{(n-1)!}{(n-4)!} \times \frac{(n-4)!}{n!} = \frac{1}{9}

  3. Step 3: Simplify the equation.

    (n−1)!n!=19\frac{(n-1)!}{n!} = \frac{1}{9}

    Since n! = n × (n-1)!, we get:

    (n−1)!n×(n−1)!=19\frac{(n-1)!}{n \times (n-1)!} = \frac{1}{9}

  4. Step 4: Solve for n.

    1n=19\frac{1}{n} = \frac{1}{9}

    Therefore, n = 9

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