Questions: UNV-103 Topic 6 DO 1 Counting and Probability Computing permutations and combinations Evaluate 10P5 and 5C4 10P5 = 5C4 =

UNV-103 Topic 6 DO 1 Counting and Probability Computing permutations and combinations Evaluate 10P5 and 5C4 10P5 = 5C4 =
Transcript text: UNV-103 Topic 6 DO 1 Counting and Probability Computing permutations and combinations Evaluate $_10P_5$ and $_5C_4$ $_10P_5 = \square$ $_5C_4 = \square$ Explanation Check
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Solution

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Solution Steps

To solve the given problem, we need to compute a permutation and a combination.

  1. For the permutation $_10P_5$, we use the formula for permutations: \( nP_k = \frac{n!}{(n-k)!} \).
  2. For the combination $_5C_4$, we use the formula for combinations: \( nC_k = \frac{n!}{k!(n-k)!} \).
Step 1: Calculate \( _{10}P_{5} \)

To compute the permutation \( _{10}P_{5} \), we use the formula: \[ _{n}P_{k} = \frac{n!}{(n-k)!} \] Substituting \( n = 10 \) and \( k = 5 \): \[ _{10}P_{5} = \frac{10!}{(10-5)!} = \frac{10!}{5!} \] Calculating this gives: \[ _{10}P_{5} = 10 \times 9 \times 8 \times 7 \times 6 = 30240 \]

Step 2: Calculate \( _{5}C_{4} \)

To compute the combination \( _{5}C_{4} \), we use the formula: \[ _{n}C_{k} = \frac{n!}{k!(n-k)!} \] Substituting \( n = 5 \) and \( k = 4 \): \[ _{5}C_{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} \] Calculating this gives: \[ _{5}C_{4} = \frac{5}{1} = 5 \]

Final Answer

Thus, the results are: \[ _{10}P_{5} = 30240 \quad \text{and} \quad _{5}C_{4} = 5 \] The final answers are: \[ \boxed{_{10}P_{5} = 30240} \] \[ \boxed{_{5}C_{4} = 5} \]

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