Question
Evaluate the expression. a) \( { }_{7} C_{5} \) b) \( { }_{7} P_{5} \) a) \( { }_{7} C_{5}=\square \) (Simplify your answer.)
Ask by Santiago Ortega. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
a) \( {}_{7} C_{5} = 21 \)
b) \( {}_{7} P_{5} = 2520 \)
Solution
1. For \( {}_{7} C_{5} \), use the combination formula:
\[
{}_{7} C_{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!}
\]
2. Simplify the factorials:
\[
\frac{7!}{5!} = 7 \times 6
\]
Therefore:
\[
{}_{7} C_{5} = \frac{7 \times 6}{2!} = \frac{42}{2} = 21
\]
3. For \( {}_{7} P_{5} \), use the permutation formula:
\[
{}_{7} P_{5} = \frac{7!}{(7-5)!} = \frac{7!}{2!}
\]
4. Calculate the factorial values:
\[
7! = 5040 \quad \text{and} \quad 2! = 2
\]
So:
\[
{}_{7} P_{5} = \frac{5040}{2} = 2520
\]
Final answers:
- a) \( {}_{7} C_{5} = 21 \)
- b) \( {}_{7} P_{5} = 2520 \)
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To evaluate the expression \( { }_{7} C_{5} \), we use the combination formula: \[ _{n} C_{r} = \frac{n!}{r!(n-r)!} \] For \( { }_{7} C_{5} \): \[ { }_{7} C_{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \] So, \( { }_{7} C_{5} = 21 \). Now, for \( { }_{7} P_{5} \) (permutation), we use the permutation formula: \[ _{n} P_{r} = \frac{n!}{(n-r)!} \] For \( { }_{7} P_{5} \): \[ { }_{7} P_{5} = \frac{7!}{(7-5)!} = \frac{7!}{2!} = \frac{7 \times 6 \times 5 \times 4 \times 3}{1} = 2520 \] Thus, \( { }_{7} P_{5} = 2520 \). Final answers: a) \( { }_{7} C_{5} = 21 \) b) \( { }_{7} P_{5} = 2520 \)
