Question

# There are 3 white, 4 blue and 1 red flowers, All of them are taken out one by one and arranged in a row in the order. How many different arrangements are possible (flowers of the same colours are similar)?.

Hint: Find the total number of arrangement of flowers. In this question it is given that 3 white flowers and 4 blue flowers are similar so we need to divide the total number of arrangements by 3! And factorial 4!. Use combinations formula $\dfrac{{n!}}{{p!q!r!}}$

Given total number of white flowers is 3
Total number of blue flowers is 4
Total number of red flower is 1
We need to find the total number of arrangements in a row such that All of them are taken out one by one and arranged in a row in the order.
The total number of ways of arranging these flowers will be 8!
However, there are 3 white flowers, 4 red flowers and 1 blue flower.
The total number of ways of arranging 3 white flowers will be 3!
The total number of ways of arranging 4 blue flowers will be 4!
Since there is a repetition of 3 and 4, the answer will be:
We have,
Total no. of different arrangements $\dfrac{{8!}}{{3!4!}}$
$= 8 \times 7 \times 5 = 280$arrangements

Note:1.the number of arrangement of a total of n objects, out of which ‘p’ are of one type, q of second type are alike, and r of a third kind are same, then such a computation is done by $\dfrac{{n!}}{{p!q!r!}}$
2. Number of ways in which n things of which r alike and the rest different can be arranged in a circle distinguishing between clockwise and anticlockwise arrangement, is $\dfrac{{\left( {n - 1} \right)!}}{r}$