Question

Smith has the work of garland making and uses different types of flowers to make garland. If he has four green flowers of rose and two red flowers of jasmine and he uses them all to make the garland, then the different types of garland he can make using the different combination of flowers is?

& A.8 \\\ & B.15 \\\ & C.16 \\\ & D.21 \\\ & \text{E}.\text{ None of these} \\\ \end{aligned}$$

Answer 1

In this question, we have four green flowers of rose and two red flowers of jasmine. From these we have to make garland. We need to find a number of different garlands that can be made. We will use the concept of permutation and combination for this. For n different things, a number of arrangements are given by n! and if out of these n items, r are of the same type and p are of other same type, then we divide the number of arrangements by $r!\times p!$. Hence we get arrangements as $\dfrac{n!}{r!p!}$. We will apply this rule of permutation here.

Complete step-by-step answer:
Here, Smith has to make a garland using four green flowers of rose and two red flowers of jasmine. We have to tell different types of garlands that can be made.
Number of green flowers of rose = 4.
Number of red flowers of jasmine = 2.
So we take total flowers as 4+2 = 6.
Therefore, we need to make garland using 6 flowers. We know, for n different items, a number of arrangements are given by n!
So the number of arrangements is 6!
But we have 4 same types of flowers and 2 other same types of flowers. Taking the same flowers will not make our garland different. So we have to reduce the number of repeated ways.
As we know, for r repeated things in n items and p type of other repeated things in n items, our arrangement becomes equal to $\dfrac{n!}{r!p!}$.
So here we have n as 6 and r as repeated green flowers of rose which is 4 and p as repeated red flowers of jasmine.
So our number of arrangements become equal to:
$\Rightarrow \dfrac{6!}{4!\times 2!}=\dfrac{6\times 5\times 4!}{4!\times 2\times 1}=\dfrac{6\times 5}{2}=3\times 5=15$.
Hence, there are 15 different ways of making a garland using four green flowers of Rose and two red flowers of jasmine.

So, the correct answer is “Option B”.

Note: Students can make the mistake of forgetting about the repeated flowers and declare this answer as 6! only. Sometimes students can make the mistake of dividing 6! by $4\times 2$ only rather than $4!\times 2!$. If there were more repeated flowers of other type s, then arrangements would be $\dfrac{n!}{r!p!s!}$ Take care while calculating factorials.