JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

20 persons are sitting in a particular arrangement around a circular table, 3 persons are to be selected for leaders. The number of ways of selection of 3 persons such that no 2 were sitting adjacent to each other is

Choose the correct answer:

Explanation

1. The General Formula

For a circular arrangement of $n$ objects, the number of ways to select $r$ objects such that no two are consecutive is given by:

\text{Number of ways} = \frac{n}{n-r} \times \binom{n-r}{r}

In this specific problem:

Total number of persons ( $n$ ) = 20
Number of persons to be selected ( $r$ ) = 3

2. Step-by-Step Calculation

Substitute the values into the formula:

\text{Ways} = \frac{20}{20-3} \times \binom{20-3}{3}

Simplify the expression:

\text{Ways} = \frac{20}{17} \times \binom{17}{3}

Now, calculate the combination $\binom{17}{3}$ :

\binom{17}{3} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1}

\binom{17}{3} = 17 \times 8 \times 5 = 680

3. Final Computation

Multiply the results:

\text{Ways} = \frac{20}{17} \times 680

\text{Ways} = 20 \times 40 = 800