Let's analyze the problem carefully.
Problem Restatement
You have:
- 3 different brown dyes
- 4 different white dyes
- some number of different purple dyes (the exact number is missing in your query)
You want to find how many combinations of dyes can be made taking at least one purple and at least one white dye.
Step 1: Clarify the number of purple dyes
Your query says "and different purple dyes," but the number is not specified. Let's denote the number of purple dyes as ppp.
Step 2: Understand the combinations
You want to form combinations of dyes such that:
- At least one white dye is chosen (from 4 white dyes)
- At least one purple dye is chosen (from ppp purple dyes)
- Brown dyes are optional (from 3 brown dyes)
Step 3: Calculate total combinations for each color group
- Brown dyes: each dye can be either chosen or not chosen
- Number of ways to choose brown dyes = 23=82^3=823=8 (including choosing none)
- White dyes: must choose at least one out of 4
- Number of ways to choose white dyes = 24−1=16−1=152^4-1=16-1=1524−1=16−1=15
- Purple dyes: must choose at least one out of ppp
- Number of ways to choose purple dyes = 2p−12^p-12p−1
Step 4: Total number of valid combinations
Since choices for brown, white, and purple are independent, multiply the number of ways:
Total combinations=(23)×(24−1)×(2p−1)=8×15×(2p−1)=120×(2p−1)\text{Total combinations}=(2^3)\times (2^4-1)\times (2^p-1)=8\times 15\times (2^p-1)=120\times (2^p-1)Total combinations=(23)×(24−1)×(2p−1)=8×15×(2p−1)=120×(2p−1)
Final answer:
Number of combinations=120×(2p−1)\boxed{ \text{Number of combinations}=120\times (2^p-1) }Number of combinations=120×(2p−1)
where ppp is the number of purple dyes. If you provide the exact number of purple dyes, I can give you the exact numeric answer!
