There are 10 seats in the first row of a theatre of which 4 are to be occupied

Question

There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is

Smart Achievers · Accepted Answer

(1)

$∴$ $x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 3 \to (remaining)$

$∴$ ${}^{3 + 5 - 1}{C_{5 - 1}} = {}^{7}{C_{4}} = {}^{7}{C_{3}}$

$∴$ $total arrangement = {}^{7}{C_{3}} \times 4! = 840$

Solution

Q.
There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is

1 840

2 600

3 276

4 640

Ans.
(1)

$∴$ $x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 3 \to (remaining)$

$∴$ ${}^{3 + 5 - 1}{C_{5 - 1}} = {}^{7}{C_{4}} = {}^{7}{C_{3}}$

$∴$ $total arrangement = {}^{7}{C_{3}} \times 4! = 840$

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