Permutation and CombinationHard
Question
Let Pn denotes the number of ways of selecting 3 people out of ′n′ sitting in a row, if no two of them are consecutive and Qn is the corresponding figure when they are in a circle. If Pn - Qn = 6, then ′n′′ is equal to:
Options
A.8
B.9
C.10
D.12
Solution
Here
∴ x1 + x2 + x3 + x4 = n - 3 ∴ x1 , x4 ≥ 0
∴ x1 + y2 + y3 + x4 ³ n – 5 x2 , x3 ≥ 1
pn = n - 5 + 4 - 1c3 = n - 2c3 y1 , y2 ≥ 0
Qn =
∴ x1 + x2 + x3 = n - 3
⇒ y1 + y2 + y3 = n - 6
Now Pn - Qn = 6
= 6
⇒ n = 10
∴ x1 + x2 + x3 + x4 = n - 3 ∴ x1 , x4 ≥ 0
∴ x1 + y2 + y3 + x4 ³ n – 5 x2 , x3 ≥ 1
pn = n - 5 + 4 - 1c3 = n - 2c3 y1 , y2 ≥ 0
Qn =
∴ x1 + x2 + x3 = n - 3
⇒ y1 + y2 + y3 = n - 6
Now Pn - Qn = 6
= 6⇒ n = 10
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