(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

For 3-letter arrangements using letters from RAINBOW, we can select any 3 letters from the 7 distinct letters available. The number of arrangements can be calculated using the formula for permutations:

P(n, r) = rac{n!}{(n - r)!}

Here, n = 7 (total letters) and r = 3 (letters to arrange):

P(7, 3) = rac{7!}{(7-3)!} = rac{7!}{4!} = 7 imes 6 imes 5 = 210

Thus, there are 210 different 3-letter arrangements.

To find the probability, we need to divide the angle of each sector by the total angle of the spinner, which is 360°:

• Red: $P(R) = \frac{72}{360} = \frac{1}{5}$
• Orange: $P(O) = \frac{30}{360} = \frac{1}{12}$
• Yellow: $P(Y) = \frac{45}{360} = \frac{1}{8}$
• Green: $P(G) = \frac{90}{360} = \frac{1}{4}$
• Blue: $P(B) = \frac{60}{360} = \frac{1}{6}$
• Indigo: $P(I) = \frac{18}{360} = \frac{1}{20}$
• Violet: $P(V) = \frac{45}{360} = \frac{1}{8}$

Including these probabilities in the table yields:

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

The expected value (E) can be calculated as:

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

Calculating each contribution:

• For Red: $€20 \times \frac{1}{5} = €4$
• For Orange: $€60 \times \frac{1}{12} = €5$
• For Yellow: $€24 \times \frac{1}{8} = €3$
• For Green: $€42 \times \frac{1}{4} = €10.5$
• For Blue: $€41 \times \frac{1}{6} = €6.8333$
• For Indigo: $€90 \times \frac{1}{20} = €4.5$
• For Violet: $€48 \times \frac{1}{8} = €6$

Summing these values:

$E(X) = 4 + 5 + 3 + 10.5 + 6.8333 + 4.5 + 6 = 31.0$

Thus, the expected value is €31.50.