Question

If Missy has 8 identical tulip plants and 4 identical daisy plants, in how many ways can she use the plants to line the walkway?

Answer 1

To answer this question, we need to know the basics of permutations and combinations. We just use the idea behind this to solve the above question. It is given that Missy has 8 identical tulip plants and 4 identical daisy plants and these can be arranged by considering the 8 identical tulips as one unit and the 4 identical daisies as another unit since they are all identical. We then use the formula for combinations to arrange n objects with a few sets of them being identical.

Complete step by step solution:
In order to solve such questions, we just need to know the basics of permutations and combinations and using these simple concepts, we can solve this question. Here, we need to use the concept of combinations since order does not matter. We use permutations in cases where order of the objects plays a role. Now we need to use this formula for combinations which is to arrange n objects where ${{p}_{1}}$ number of objects are of one kind, ${{p}_{2}}$ objects are of second kind, ……. ${{p}_{k}}$ objects are of ${{k}^{th}}$ kind.
$= \dfrac{n!}{{{p}_{1}}!.{{p}_{2}}!.{{p}_{3}}!\ldots {{p}_{k}}!}$
For the given question, the total number of flowers to be arranged is n. 8 of them are identical and are tulips. Let us take ${{p}_{1}}=8.$ The other 4 of them are daisies and let us take them as ${{p}_{2}}=4.$
We plug these values in the formula and simplify.
$= \dfrac{12!}{8!.4!}$
We know that $n!$ is given as,
$n!=n.\left( n-1 \right).\left( n-2 \right).......3.2.1$
Expanding the 12 factorial up to 8 factorial so that they can be cancelled out as they are same,
$= \dfrac{12.11.10.9.8!}{8!.4!}$
Cancelling the 8 factorial in the numerator and denominator and expanding 4 factorial,
$= \dfrac{12.11.10.9}{4.3.2.1}$
Cancelling the terms which have common factors,
$= 11.5.9$
Taking the product of these terms,
$= 495$
Hence, Missy can arrange these 8 identical tulip plants and 4 identical daisy plants to line the walkway in 495 ways.

Note: It is essential to know the basic concepts of permutations and combinations in order to solve this. Another way to solve this problem is without using the formula. Assume that the total 12 plants are unique, we can arrange them in $12!$ ways. Now since 8 of them are identical and the other 4 are identical, it does not matter in what order these 8 tulips or 4 daisies are arranged among themselves. Hence the ways in which these are arranged will be $8!.4!$ . We divide the total number of ways by these identical cases to obtain the answer.