Answer these questions based on the following information.
There are 2 white candles, 3 red candles, 4 green candles and 3 yellow candles in a brand new packet. All the candles are uniform in shape and size, except their colour, as told above.
Find the number of ways of selecting at least one candle.

Options

A. A) `2^22-1`
B. B) 239
C. C) 72
D. D) 12

Answer

To find the number of ways of selecting at least one candle from the given candles, we can follow these steps:

### Step 1: Identify the number of each type of candle
- White candles: 2
- Red candles: 3
- Green candles: 4
- Yellow candles: 3

### Step 2: Use the formula for selecting items
The formula for finding the number of ways to select at least one item from several groups of identical items is given by:
\[ (P + 1)(Q + 1)(R + 1)(S + 1) - 1 \]
Where \(P\), \(Q\), \(R\), and \(S\) are the counts of each type of candle.

### Step 3: Substitute the values into the formula
Here, we have:
- \(P = 2\) (white candles)
- \(Q = 3\) (red candles)
- \(R = 4\) (green candles)
- \(S = 3\) (yellow candles)

Now substituting these values into the formula:
\[ (2 + 1)(3 + 1)(4 + 1)(3 + 1) - 1 \]

### Step 4: Calculate each term
Calculating each term:
- \(2 + 1 = 3\)
- \(3 + 1 = 4\)
- \(4 + 1 = 5\)
- \(3 + 1 = 4\)

### Step 5: Multiply the results
Now we multiply these results:
\[ 3 \times 4 \times 5 \times 4 \]

Calculating step-by-step:
1. \(3 \times 4 = 12\)
2. \(12 \times 5 = 60\)
3. \(60 \times 4 = 240\)

### Step 6: Subtract 1
Now we subtract 1 to account for the case where no candle is selected:
\[ 240 - 1 = 239 \]

### Final Answer
Thus, the number of ways of selecting at least one candle is **239**.

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