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The problem asks how many distinct orders of light bulbs can be created when Tammy arranges 11 light bulbs, with the following breakdown:

- 3 yellow bulbs
- 4 white bulbs
- 4 pink bulbs

Since bulbs of the same color are considered identical, this is a permutation problem where we need to account for identical objects.

The formula to find the number of distinct permutations when some objects are identical is:

\[
\frac{n!}{k_1! k_2! \dots k_m!}
\]

where \(n\) is the total number of objects, and \(k_1, k_2, \dots, k_m\) are the counts of the identical objects.

In this case:

- \(n = 11\) (total bulbs),
- \(k_1 = 3\) (yellow bulbs),
- \(k_2 = 4\) (white bulbs),
- \(k_3 = 4\) (pink bulbs).

The number of distinct orders is:

\[
\frac{11!}{3! 4! 4!}
\]

Let's compute this step by step.

First, calculate the factorials:

\[
11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39,916,800
\]
\[
3! = 3 \times 2 \times 1 = 6
\]
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

Now, substitute these values into the formula:

\[
\frac{11!}{3! 4! 4!} = \frac{39,916,800}{6 \times 24 \times 24} = \frac{39,916,800}{3,456} = 11,550
\]

So, the number of distinct orders of light bulbs is **11,550**.

Would you like further details or have any questions? Here are 5 related questions you might find interesting:

1. How would the number of arrangements change if Tammy added another color of bulbs?
2. What if all 11 bulbs were unique (no identical bulbs)?
3. How can this problem be generalized to any number of colors and bulbs?
4. What is the smallest number of bulbs where identical colors significantly affect the total permutations?
5. How does this concept of permutations apply to card shuffling or seating arrangements?

**Tip:** When dealing with identical objects in permutations, always account for each group by dividing by their factorial to avoid overcounting!

Tammy is putting 11 colored light bulbs into a string of lights. There are 3 yellow light bulbs, 4 white light bulbs, and 4 pink light bulbs. How many distinct orders of light bulbs are there if two light bulbs of the same color are considered identical (not distinct)?

Learn how to solve a combinatorics problem involving 11 colored light bulbs. Calculate the distinct arrangements of 3 yellow, 4 white, and 4 pink bulbs using factorials. This problem is suitable for students in Grades 9-12.

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