Question

There are 20 persons seated round a circular table. The no. of ways can 4 persons be selected so that there are no neighbourers.

• A. $17C_{4} -^{15}C_{2}$
• B. $17C_{4}$
• C. $17C_{4}.^{15}C_{2}$
• D. $17C_{4} -^{15}$

Answer 1

Answer: (C)

$17C_{4}.^{15}C_{2}$

Answer 2

Let us assume that the persons 1, 2, 3, ...... 20 are in a row. Let the selected persons be denoted by x and other by y.

Then the no. of ways of selecting 4 persons so that no two are neighbourers is same as no. of ways of arranging 4 x's and 16 y's in a row so that no. two x's are together. This can be done in ¹⁷C₄ ways.

Since the 20 persons are seated round a circular table, selection of 1 and 20 together is not allowed. In the above selections, the no. of ways containing 1 and 20 = No. of ways of selecting 2 persons from 3, 4, 5, 6 ..... 18 without neighbourers.

This can be done in ¹⁵C₂ ways.

Hence the required no. of ways = $17C_{4} -^{15}C_{2}$.

Method II: Convert circular arrangement into linear. Arrangement by selecting 1 person.

No. of ways = $\frac{20{C_{1}}^{17 - 3 + 1}C_{3}}{4} = \frac{20{C_{1}}^{15}C_{3}}{4}$.