First slide

Combinations

Question

A table has provision for 7 seats, 4 being on one side facing the window and 3 being on the opposite side. The number of ways in which 7 people can be seated at the table if 2 people, X and Y, must sit on the same side, is

Moderate

A

3260
B

2160
C

3350
D

None of these

Solution

When X and Y sit on the side facing the window—
First, we have to select 2 persons for the side facing the window and 3 for the opposite side from the remaining 5. The number of selections for this is equal to $^{5} C_{2} \times^{3} C_{3}$ .
For each of the selections, the number of arrangements = 4! × 3!.
∴ The number of ways to sit when X, Y are on the side facing the window $=^{5} C_{2} \times^{3} C_{3} \times 4! \times 3!$ .
When X and Y sit on the other side
As before, the number of ways to sit
$=^{5} C_{1} \times^{4} C_{4} \times 4! \times 3!$
∴ Required number of ways
$\begin{array}{l} =^{5} C_{2} \times^{3} C_{3} \times 4! \times 3! +^{5} C_{1} \times^{4} C_{4} \times 4! \times 3! \\ = \frac{5 \times 4}{2} \times 1 \times 24 \times 6 + 5 \times 1 \times 24 \times 6 = 2160 \end{array}$

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