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Question: In how many ways, we can arrange \(3\) red flowers, \(4\) yellow flowers, and \(5\) white flowers in...

In how many ways, we can arrange 33 red flowers, 44 yellow flowers, and 55 white flowers in a row? In how many ways, this is possible if the white flowers are to be separated in any arrangement? (Flowers of the same color are identical)

Explanation

Solution

Hint : Find the total number of flowers and then the number of arrangement using the formula for non-identical objects, which is given as; when there are n objects, out of which n1{n_1} are identical, n2{n_2} are identical and so on, then the number of arrangement is given as:
Number of arrangements=n!n1!×n2!×…×nk! = \dfrac{{n!}}{{{n_1}! \times {n_2}! \times \ldots \times {n_k}!}}

Complete step by step solution :
We have given that there are 33 red flowers, 44 yellow flowers, and 55 white flowers.
The goal is to find the number of ways that we can arrange these flowers in a row and then find the number of ways if the white flowers are separated in any arrangement.
We have given flowers,
33red flowers, 44 yellow flowers, and 55 white flowers.
Then the total numbers of flower are:
Total flowers=3+4+5=12 = 3 + 4 + 5 = 12flowers
Then the number of ways to arrange these flowers in a row is given as:
Number of arrangement=12!3! ×4! ×5! = \dfrac{{12!}}{{3!{\text{ }} \times 4!{\text{ }} \times 5!}}
Simplify the above fraction to get the number of arrangements of flowers in a row.
Number of arrangement=12×11×10×9×8×7×6×5!3×2×4×3×2×5! = \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5!}}{{3 \times 2 \times 4 \times 3 \times 2 \times 5!}}
Number of arrangement=12×11×10×9×8×7×63×2×4×3×2 = \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6}}{{3 \times 2 \times 4 \times 3 \times 2}}
Number of arrangement=27720 = 27720
So, there are2772027720ways to arrange 1212flowers in a row.
For getting the solution of the second part, think about the number of ways to arrange the 33 red and 4 yellow flowers in a row.
Then, the total numbers of flowers are(3+4=7)\left( {3 + 4 = 7} \right).
So, the number of arrangement of 33 red and 4 yellow flowers in a row can be given as:
Number of arrangement=7!3! ×4! = \dfrac{{7!}}{{3!{\text{ }} \times 4{\text{!}}}}
Simplify the above fraction:
Number of arrangement=7×6×5×4!3×2×4! = \dfrac{{7 \times 6 \times 5 \times 4!}}{{3 \times 2 \times 4{\text{!}}}}
Number of arrangement=7×5 = 7 \times 5
Number of arrangement=35 = 35
Therefore, there are 35 ways to arrange the 33 red and 4 yellow flowers in a row.
Now, assume 7 places for 33 red and 4 yellow flowers and one place for arranging a white flower, and then the total places are 8. Now, we have to select 5 places out of 8 places. So, the choice of 5 places out of 8 is given as:
5C8=8!5!(8−5)!^5{C_8} = \dfrac{{8!}}{{5!\left( {8 - 5} \right)!}}
5C8=8×7×6×5!5! ×3!^5{C_8} = \dfrac{{8 \times 7 \times 6 \times 5!}}{{5!{\text{ }} \times 3!}}
5C8=8×7×63×2^5{C_8} = \dfrac{{8 \times 7 \times 6}}{{3 \times 2}}
5C8=56^5{C_8} = 56
So, the total number of arrangements is given as:
Number of ways=35×56 = 35 \times 56
Number of ways=1960 = 1960
So, the required numbers of ways are 1960.1960.

Note : The given red, yellow and white flowers are identical, so if we use them in place of each other it does not make any changes, therefore while finding the total ways to arrange the flowers, we have to take out the repetition count of arrangements out of the total arrangements and thus the permutation has the form:
P=7!3! ×4! ×5!P = \dfrac{{7!}}{{3!{\text{ }} \times 4!{\text{ }} \times 5!}}