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Question

There are 12 seats in a row of a cinema theatre which is nearer to the entrance. 4 persons enter the theatre and occupy the seats in that row (Before their entry the seats are vacant). If n1is the number of ways in which they occupy the seats, such that no two persons are together and n2 is the number of ways in which each person has exactly one neighbour n1:n2 is

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Solution

The correct option is **D** 7:2

(1) There are8 empty seats and 9 gaps.

∴n1=9P4=3024

(2) Each person has exactly one neighbour.

The 4 persons can be paired like P1P2:P3P4 (say)

Letx1 be the number of empty seats before P1P2 and

x2 be the number of empty seats between P1P2 and P3P4.

Letx3 be the number of empty seats after P3P4.

x1+x2+x3=8 with x1≥0,x2≥1,x3≥0

Put y1=xt;y2=x2−1;y3=x3

Number of solutions of y1+y2+y3=7 each y1≥0

=9C2

For each pair 2 can be selected in 4C2 ways, each pair can be arranged in ⌊2 ways.

∴ n2=9C42C2 ⌊2⌊2=864

∴ n1n2=3024864=72

(1) There are8 empty seats and 9 gaps.

∴n1=9P4=3024

(2) Each person has exactly one neighbour.

The 4 persons can be paired like P1P2:P3P4 (say)

Letx1 be the number of empty seats before P1P2 and

x2 be the number of empty seats between P1P2 and P3P4.

Letx3 be the number of empty seats after P3P4.

x1+x2+x3=8 with x1≥0,x2≥1,x3≥0

Put y1=xt;y2=x2−1;y3=x3

Number of solutions of y1+y2+y3=7 each y1≥0

=9C2

For each pair 2 can be selected in 4C2 ways, each pair can be arranged in ⌊2 ways.

∴ n2=9C42C2 ⌊2⌊2=864

∴ n1n2=3024864=72

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