Question

Prove that the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated is $\dfrac{1}{2}4!4!$.

Answer 1

Hint: In permutations and combinations, we have a formula to calculate the number of circular permutations of ‘n’ different things on the garland. This number of permutations is equal to $\dfrac{1}{2}\left( n-1 \right)!$. Use this formula to solve this question. Also, since 4 particular flowers are never separated, solve this question by considering these 4 flowers as a single unit.

Complete step-by-step answer:
Before proceeding with the question, we must know all the formulas that will be required to solve this question.
In permutations and combinations, we have a formula that can be used to calculate the number of arrangements of n different things on the garland. The number of arrangements of n different things on the garland is equal to $\dfrac{1}{2}\left( n-1 \right)!...........\left( 1 \right)$.
Also, the number of ways in which n things can be arranged within themselves is equal to $n!............\left( 2 \right)$.
In this question, we are required to find the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated. Let us consider those 4 particular flowers as a single unit. If we consider these 4 particular flowers as a single unit, then we have to arrange 5 things (1 single unit of 4 particular flowers and other 4 flowers) instead of 8 things on the garland. Using formula $\left( 1 \right)$, the number of arrangements of these 5 elements is equal to $\dfrac{1}{2}\left( 5-1 \right)!=\dfrac{1}{2}4!............\left( 3 \right)$.
Since those 4 flowers that we had considered as a single unit were different from each other, so they can be arranged within themselves. Using formula $\left( 2 \right)$, the number of ways in which we can arrange these 4 flowers within themselves is equal to 4!$...........\left( 4 \right)$
Multiplying the numbers obtained in $\left( 3 \right)$ and $\left( 4 \right)$, the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated is equal to $\dfrac{1}{2}4!4!$.
Hence proved.

Note: There is a possibility that one may forget to arrange the 4 particular flowers among themselves which we have considered as a single unit. But since those four flowers are different from each other, it is necessary to arrange them within themselves.