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in Complex number and Quadratic equations by (46.6k points)

Ten persons are arranged in a row. The number of ways of selecting four persons so that no two persons sitting next to each other are selected is 

(A)  34 

(B)  36 

(C)  35 

(D)  None of these 

1 Answer

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Best answer

Correct option (C) 35

To each selection of 4 persons we associate binary sequence of the form 1001001010 where 1(0) at ith place means the ith person is selected (not selected).

There exists one-to-one correspondence between the set of selections of 4 persons and set of binary sequence containing 6 zeros and 4 ones.

We are interested in the binary sequences in which no 2 ones are consecutive. We first arrange 6 zeros:

0 0 0 0 0 0 

This can be done in just one way.

Now, 4 ones can be arranged at any of the 4 places marked with a cross in the following arrangement:

x 0 x 0 x 0 x 0 x 0 x 0 

We can arrange 4 ones at 7 places in 7C4 = 35 ways. 

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