# 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

Verified

362.4k+ views

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.

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.

Last updated date: 24th Sep 2023

•

Total views: 362.4k

•

Views today: 10.62k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

What is the basic unit of classification class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers