Combination Problem: Selecting 6 Roses of 3 Colors

[cyeokpeng]

I got this problem, don't know how to solve.

Of 10 variety of roses, 3 are pink, 5 are red and 2 are yellow. Calculate the number of ways in which we can select 6 roses, so that in the selection, at least one rose of each color is included.

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[Tide]

What, exactly, have you tried so far?

 

[Architeuthis Dux]

I would approach this problem by using the principles of combinatorics. In this case, we are dealing with a combination problem where order does not matter and repetition is not allowed. This means we will be using the combination formula, nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items we are selecting.

In this problem, n = 10 (total number of roses) and r = 6 (number of roses we want to select). We also know that we need at least one rose of each color, so we can set aside one pink, one red, and one yellow rose, leaving us with 7 remaining roses to choose from.

Using the combination formula, we can calculate the number of ways to select 6 roses with at least one of each color as:

(7C3) * (4C2) * (2C1) = (7! / (3!(7-3)!)) * (4! / (2!(4-2)!)) * (2! / (1!(2-1)!)) = 35 * 6 * 2 = 420

Therefore, there are 420 different ways to select 6 roses from 10 varieties with at least one of each color included. I hope this helps you solve the problem!