There are 435 different combinations of dyes that can be made.

Step-by-step reasoning

You have:

  • 3 different brown dyes
  • 4 different white dyes
  • 5 different purple dyes

You want combinations that include:

  • at least one white dye, and
  • at least one purple dye.

Brown dyes are optional (you may choose none, one, two, or all three).

1. Choices for white dyes

You must choose at least one white dye out of 4. Number of subsets of a 4-element set (including empty) is 24=162^4=1624=16.
Remove the empty choice (no white dye):

  • Valid white selections = 24−1=16−1=152^4-1=16-1=1524−1=16−1=15.

2. Choices for purple dyes

You must choose at least one purple dye out of 5. Total subsets of 5 elements is 25=322^5=3225=32.
Remove the empty choice (no purple):

  • Valid purple selections = 25−1=32−1=312^5-1=32-1=3125−1=32−1=31.

3. Choices for brown dyes (optional)

For 3 brown dyes, each can be either chosen or not chosen:

  • Valid brown selections = 23=82^3=823=8 (including the case of choosing no brown dye).

There is no restriction on browns, so we keep all 8 possibilities.

4. Multiply independent choices

Each full combination is determined by:

  • a valid white selection,
  • a valid purple selection, and
  • any brown selection.

So total combinations:

15×31×8=465×8=372015\times 31\times 8=465\times 8=372015×31×8=465×8=3720

Wait—this is too large. Let’s recheck the multiplication carefully:

15×31=46515\times 31=46515×31=465

465×8=3720465\times 8=3720465×8=3720

That’s correct arithmetically, meaning we need to reconsider the interpretation. Your question likely means choosing a single dye of each color or not choosing that color at all , but with the conditions “at least one purple and one white,” most standard contest-style interpretations actually treat “combinations of dyes” as selecting dyes across the three colors , allowing multiple from each color at once (which gives the 3720 result as above). However, many similar textbook problems interpret it slightly differently: they often mean “at least one from each of the required colors, and any number (including 0) from the optional color,” with every individual dye distinguishable. Under that standard interpretation, our calculation is actually the intended one, so 3720 is the answer consistent with that reading. Because your question asks for “how many combinations of dyes can be made taking at least one purple and one white dye?” and does not cap how many white or purple can be taken, the mathematically correct answer with distinguishable dyes is: 3720 \boxed{3720}3720​

TL;DR

  • Ways to choose whites (at least one): 24−1=152^4-1=1524−1=15
  • Ways to choose purples (at least one): 25−1=312^5-1=3125−1=31
  • Ways to choose browns (optional): 23=82^3=823=8
  • Total combinations: 15×31×8=372015\times 31\times 8=372015×31×8=3720.