Question

1. If nP4 = 14 n-2P3; find the value of n.

Answer 1

SOLUTION

GIVEN

$\sf{ \: {}^{n} P_4 = 14 \times {}^{n - 2} P_3}$

TO DETERMINE

The value of n,

EVALUATION

Here it is given that

$\sf{ \: {}^{n} P_4 = 14 \times {}^{n - 2} P_3}$

$\implies \displaystyle \sf{ \frac{ n\: !}{( n - 4)\: !} = 14 \times \frac{ (n - 2)\: !}{( n - 5)\: !} }$

$\implies \displaystyle \sf{n(n - 1)(n - 2)(n - 3) = 14 \times (n - 2)(n - 3)(n - 4) }$

$\implies \displaystyle \sf{n(n - 1) = 14(n - 4) }$

$\implies \displaystyle \sf{ {n}^{2} - n= 14n - 56 }$

$\implies \displaystyle \sf{ {n}^{2} - 15n + 56 = 0 }$

$\implies \displaystyle \sf{ {n}^{2} - 7n - 8n+ 56 = 0 }$

$\implies \displaystyle \sf{n(n - 7) - 8(n - 7) = 0 }$

$\implies \displaystyle \sf{(n - 7) (n - 8) = 0 }$

$\implies \displaystyle \sf{n = 7 \: ,\: 8 }$

FINAL ANSWER

The required value of n is 7 , 8

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