(a) (i) Find the number of different arrangements that can be made using all the letters of the word RAINBOW - Leaving Cert Mathematics - Question 3 - 2018

To find the number of different 3-letter arrangements, we use the combination of selecting 3 letters from the 7 available, followed by permuting them.

First, the number of ways to choose 3 letters from 7 can be expressed as:

inom{7}{3} = rac{7!}{3!(7-3)!} = 35

For each selection, the letters can be arranged in:

$3! = 6$

Therefore, the total number of different 3-letter arrangements is:

$35 \times 6 = 210$

Thus, the number of different 3-letter arrangements is 210.

The probability of landing on each sector is calculated as the ratio of the angle of the sector to the total angle of the spinner (360°). Calculating for each sector:

• Red: 72°: ( \frac{72}{360} = \frac{1}{5} )
• Orange: 30°: ( \frac{30}{360} = \frac{1}{12} )
• Yellow: 45°: ( \frac{45}{360} = \frac{1}{8} )
• Green: 90°: ( \frac{90}{360} = \frac{1}{4} )
• Blue: 60°: ( \frac{60}{360} = \frac{1}{6} )
• Indigo: 18°: ( \frac{18}{360} = \frac{1}{20} )
• Violet: 45°: ( \frac{45}{360} = \frac{1}{8} )

So, the completed table is:

Colour	Angle	Probability	Prize
Red	72°	1/5	€20
Orange	30°	1/12	€60
Yellow	45°	1/8	€24
Green	90°	1/4	€8
Blue	60°	1/6	€42
Indigo	18°	1/20	€90
Violet	45°	1/8	€48

The expected value (E) can be calculated as follows:

$E(X) = \sum_{i=1}^{n} (X_i \cdot P(X_i))$

Where:

• $X_i$ is the prize for each color,
• $P(X_i)$ is the probability for each color.

Plugging the values into the formula:

$E(X) = (20 \cdot \frac{1}{5}) + (60 \cdot \frac{1}{12}) + (24 \cdot \frac{1}{8}) + (8 \cdot \frac{1}{4}) + (42 \cdot \frac{1}{6}) + (90 \cdot \frac{1}{20}) + (48 \cdot \frac{1}{8})$

Calculating:

• $20 \cdot \frac{1}{5} = 4$
• $60 \cdot \frac{1}{12} = 5$
• $24 \cdot \frac{1}{8} = 3$
• $8 \cdot \frac{1}{4} = 2$
• $42 \cdot \frac{1}{6} = 7$
• $90 \cdot \frac{1}{20} = 4.5$
• $48 \cdot \frac{1}{8} = 6$

Summing these values gives:

$E(X) = 4 + 5 + 3 + 2 + 7 + 4.5 + 6 = 31.50$

Therefore, the expected value of the prize is €31.50.