Question

Find the number of ways of arranging 6 red roses and 3 yellow roses of different sizes into a garland. In how many of them (i) all the yellow roses are together (ii) no two yellow roses are together

[Hint : The number of circular permutations like the garlands of flowers, chains of beads etc., of n things = 1/2(n – 1)!]

Answer 1

Total number of roses = 6 + 3 = 9

∴ The number of ways of arranging 6 red roses and 3 yellow roses of different sizes into a garland = 1/2(9 – 1)! = 1/2 × 8!

= 1/2 × 40,320 = 20,160

(i) Treat all the 3 yellow roses as one unit. Then we have 6 red roses and one unit of yellow roses. They can be arranged in garland form in (7 – 1)! = 6! ways.

Now, the 3 yellow roses can be arranged among themselves in 3! ways. But in the case of garlands, clockwise arrangements look alike.

∴ The number of required arrangements = 1/2 × 6! × 3!

= 1/2 × 720 × 6 = 2160

(ii) First arrange the 6 red roses in garland form in 5! ways. Then we can find 6 gaps between them. The 3 yellow roses can be arranged in these 6 gaps in ⁶P₃ ways.

But in the case of garlands, clock–wise and anti–clockwise arrangements look alike.

∴ The number of required arrangements

= 1/2 × 5! × ⁶P₃

= 1/2 × 120 × 6 × 5 × 4 = 7200