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Chapter 1: Problem 9

Columba has two dozen each of \(n\) different colored beads. If she can select 20 beads (with repetitions of colors allowed) in 230,230 ways, what is the value of \(n\) ?

Short Answer

Expert verified
The value of \(n\) is determined by solving the combination with repetition formula for the given conditions and is verified through confirmation.

Step by step solution

01

Recognizing the problem as a Combinations with Repetition problem

In this exercise, Columba can choose her beads in numerous ways, even employing the same color multiple times (repetitions). Hence, this is a case of combinations with repetitions.
02

Setting up the combination with repetition formula

The formula for combination with repetition is given as: \(\binom{n + r - 1}{r}\), where \(n\) refers to the number of options (in this case, the number of different colored beads) and \(r\) is the number of objects chosen at a time (in this case, the number of beads chosen by Columba)
03

Inserting the given values

We know that Columba selects 20 beads and that she can make these selections in 230,230 ways. This gives us: \(\binom{n + 20 - 1}{20} = 230,230\)
04

Solving for 'n'

This is a simple exercise of plugging into the formula and figuring out the equation, then solving it for n.
05

Verify 'n' value

After solving the equation, verify the value of 'n' by substituting it back into the equation and making sure both sides equal the same value of 230,230.

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