Four letters are selected at random from the 8 letters of the word TOMORROW. Find the probability that the selection contains at least one $$\(\mathrm{O}\)$$ and at least one $$\(\mathrm{R}\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_s21_qp_52 Year：2021 Question No：6(c)
Answer:
Method 1 Identified scenarios
OORR \(\quad{ }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{2} \times\left[{ }^{3} \mathrm{C}_{0}\right]=3 \times 1=3\)
ORR_ \(\quad{ }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2} \times{ }^{3} \mathrm{C}_{1}=3 \times 1 \times 3=9\)
OOR_ \(\quad{ }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1} \times{ }^{3} \mathrm{C}_{1}=3 \times 2 \times 3=18\)
OR_ \( { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1} \times{ }^{3} \mathrm{C}_{2}=3 \times 2 \times 3=18 \)
OOOR \({ }^{3} \mathrm{C}_{3} \times{ }^{2} \mathrm{C}_{1} \times\left[{ }^{3} \mathrm{C}_{0}\right]=1 \times 2=2\)
Total 50
Probability \(=\frac{50}{{ }^{8} C_{4}}\)
\(\frac{50}{70}\) or \(0.714\)
Method 2 Identified outcomes
\(\begin{array}{ll}\text { ORTM } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { ORTW } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { ORMW } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { ORRM } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { ORRW } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { ORRT } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { OROR } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { OROT } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { OROM } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { OROW } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { OROO } & { }^{3} \mathrm{C}_{3} \times{ }^{2} \mathrm{C}_{1}=2\end{array}\)
Total 50
Probability \(=\frac{50}{{ }^{8} C_{4}}\)
\(\frac{50}{70}\) or \(0.714\)
OORR \(\quad{ }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{2} \times\left[{ }^{3} \mathrm{C}_{0}\right]=3 \times 1=3\)
ORR_ \(\quad{ }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2} \times{ }^{3} \mathrm{C}_{1}=3 \times 1 \times 3=9\)
OOR_ \(\quad{ }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1} \times{ }^{3} \mathrm{C}_{1}=3 \times 2 \times 3=18\)
OR_ \( { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1} \times{ }^{3} \mathrm{C}_{2}=3 \times 2 \times 3=18 \)
OOOR \({ }^{3} \mathrm{C}_{3} \times{ }^{2} \mathrm{C}_{1} \times\left[{ }^{3} \mathrm{C}_{0}\right]=1 \times 2=2\)
Total 50
Probability \(=\frac{50}{{ }^{8} C_{4}}\)
\(\frac{50}{70}\) or \(0.714\)
Method 2 Identified outcomes
\(\begin{array}{ll}\text { ORTM } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { ORTW } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { ORMW } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { ORRM } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { ORRW } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { ORRT } & { }^{3} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { OROR } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{2}=3 \\ \text { OROT } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { OROM } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { OROW } & { }^{3} \mathrm{C}_{2} \times{ }^{2} \mathrm{C}_{1}=6 \\ \text { OROO } & { }^{3} \mathrm{C}_{3} \times{ }^{2} \mathrm{C}_{1}=2\end{array}\)
Total 50
Probability \(=\frac{50}{{ }^{8} C_{4}}\)
\(\frac{50}{70}\) or \(0.714\)
Knowledge points:
5.2.1 understand the terms permutation and combination, and solve simple problems involving selections
Solution:
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