The yearbook committee has to pick a colour scheme for this year's yearbook. There are 7 colour to choose from (red, orange, yellow, green, blue, indigo, and violet). How many different colour shemes are possible if the committee can select at most 2 colors?

Answer

To solve the problem of how many different color schemes are possible if the committee can select at most 2 colors from 7 available colors (red, orange, yellow, green, blue, indigo, and violet), we can break it down into two cases: selecting one color and selecting two colors.

### Step-by-Step Solution:

**Step 1: Count the number of ways to select one color.**
- There are 7 colors available.
- If the committee selects only one color, the number of ways to choose one color from seven is simply 7.

**Step 2: Count the number of ways to select two colors.**
- The committee can choose 2 colors from the 7 available colors.
- Since the order of selection does not matter, we use combinations to find the number of ways to choose 2 colors from 7.
- The number of combinations can be calculated using the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
- Here, \( n = 7 \) and \( r = 2 \):
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2! \cdot 5!} \]
- Simplifying this:
\[ = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

**Step 3: Calculate the total number of color schemes.**
- Now, we combine the results from both cases:
- From Step 1, we have 7 ways to choose one color.
- From Step 2, we have 21 ways to choose two colors.
- Therefore, the total number of different color schemes is:
\[ 7 + 21 = 28 \]

### Final Answer:
The total number of different color schemes possible is **28**.

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