Question

Find the number of ways in which 6 red roses and 3 white roses (all roses different) can form a garland such that all the white roses come together

Answer 1

Here in this problem we use the formula for circular permutation.

Complete step by step solution:
There are 6 red roses and 3 white roses (all roses different) which need to be formed into a garland.
The formula for circular permutation of n distinct objects into an arrangement which can be flipped is $\dfrac{{\left( {n - 1} \right)!}}{2}$. So considering the three white roses as one element, and the six red roses, there are seven distinct elements.
Hence the number of permutations is $\dfrac{{\left( {7 - 1} \right)!}}{2}$.
Again, the white roses also arrange amongst themselves, so, the total number of arrangements is

\dfrac{{\left( {7 - 1} \right)!}}{2}\left( {3!} \right)\\\ = \dfrac{{6!}}{2}\left( {3!} \right)\\\ = 6! \times 3\\\ = 2160 \end{array}$$ **Note:** In these types of questions it is important to understand that it is a case of circular permutation and not linear permutation.