Four letters are selected at random from the 8 letters of the word TOMORROW.
Find the probability that the selection contains at least one O and at least one R.
Solution
The word TOMORROW consists of the letters T, O, M, O, R, R, O, W.
We need to find the probability that a selection of 4 letters contains at least one O and at least one R.
Method 1: Identified scenarios
- OORR: \(\binom{3}{2} \times \binom{2}{2} \times \binom{3}{0} = 3 \times 1 = 3\)
- ORR_: \(\binom{3}{1} \times \binom{2}{2} \times \binom{3}{1} = 3 \times 1 \times 3 = 9\)
- OOR_: \(\binom{3}{2} \times \binom{2}{1} \times \binom{3}{1} = 3 \times 2 \times 3 = 18\)
- OR__: \(\binom{3}{1} \times \binom{2}{1} \times \binom{3}{2} = 3 \times 2 \times 3 = 18\)
- OOOR: \(\binom{3}{3} \times \binom{2}{1} \times \binom{3}{0} = 1 \times 2 = 2\)
\(Total = 50\)
Probability = \(\frac{50}{\binom{8}{4}} = \frac{50}{70}\)
Method 2: Identified outcomes
- ORTM: \(\binom{3}{1} \times \binom{2}{1} = 6\)
- ORTW: \(\binom{3}{1} \times \binom{2}{1} = 6\)
- ORMW: \(\binom{3}{1} \times \binom{2}{1} = 6\)
- ORRM: \(\binom{3}{1} \times \binom{2}{2} = 3\)
- ORRW: \(\binom{3}{1} \times \binom{2}{2} = 3\)
- ORRT: \(\binom{3}{1} \times \binom{2}{2} = 3\)
- OROR: \(\binom{3}{2} \times \binom{2}{1} = 6\)
- OROT: \(\binom{3}{2} \times \binom{2}{1} = 6\)
- OROM: \(\binom{3}{2} \times \binom{2}{1} = 6\)
- OROW: \(\binom{3}{2} \times \binom{2}{1} = 6\)
- OROO: \(\binom{3}{2} \times \binom{2}{1} = 6\)
\(Total = 50\)
Probability = \(\frac{50}{\binom{8}{4}} = \frac{50}{70}\)
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