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

For the 3-letter arrangements, we can choose any 3 letters from the 7 letters available. The number of combinations of choosing 3 letters from 7 is given by:

$C(7,3) = \frac{n!}{r!(n-r)!} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!\times 4!} = 35$

Once we select any 3 letters, the permutations for every selection of 3 letters would be:

$3! = 6$

Thus, the total number of 3-letter arrangements is:

$C(7,3) \times 3! = 35 \times 6 = 210$

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

The probability for each sector can be calculated by dividing the angle of each color by the total angle of the spinner. The total angle of a circle is 360°. Hence,

• For Red: $\frac{72}{360} = \frac{1}{5}$
• For Orange: $\frac{30}{360} = \frac{1}{12}$
• For Yellow: Given as $\frac{1}{8}$.
• For Green: $\frac{90}{360} = \frac{1}{4}$
• For Blue: $\frac{60}{360} = \frac{1}{6}$
• For Indigo: $\frac{18}{360} = \frac{1}{20}$
• For Violet: $\frac{45}{360} = \frac{1}{8}$

Thus, 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 using the formula:

$E(X) = \sum (X_i \cdot P(X_i))$

Where $X_i$ represents the cash prize and $P(X_i)$ is the probability of that prize being won. We calculate:

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

Calculating each term gives:

• Red: $20 \cdot \frac{1}{5} = 4$
• Orange: $60 \cdot \frac{1}{12} = 5$
• Yellow: $24 \cdot \frac{1}{8} = 3$
• Green: $8 \cdot \frac{1}{4} = 2$
• Blue: $42 \cdot \frac{1}{6} = 7$
• Indigo: $90 \cdot \frac{1}{20} = 4.5$
• Violet: $48 \cdot \frac{1}{8} = 6$

Summing these we get:

$E(X) = 4 + 5 + 3 + 2 + 7 + 4.5 + 6 = 31.5$

Thus, the expected value of the prize is €31.5.