If n+5Pn+1=1/2(11)(n-1)(n+3Pn) then n is equal to what?
Understand the Problem
The question pertains to mathematical problems involving combinatorics and functions, specifically focusing on solving equations related to permutations and combinations. The examples provide step-by-step solutions to each problem, guiding on how to derive values for the variables involved.
Answer
$n = 3$
Answer for screen readers
The value of (n) is 3.
Steps to Solve
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Understanding the Equation The equation given in Example 2 is $$ n^3 + 2C_3 = n^3 + 3P_2 = 20 $$ We need to interpret (3P_2) and convert it into a suitable form.
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Expanding (3P_2) Using the permutation formula, (rP_k = \frac{r!}{(r-k)!}), we calculate: $$ 3P_2 = \frac{3!}{(3-2)!} = \frac{3!}{1!} = 3 \times 2 = 6 $$
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Substituting into the Equation Replace (3P_2) in the original equation: $$ n^3 + 6 = 20 $$
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Solving for (n^3) We can isolate (n^3) by subtracting 6 from both sides: $$ n^3 = 20 - 6 = 14 $$
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Finding the Value of (n) To solve for (n), we take the cube root: $$ n = \sqrt[3]{14} $$
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Determining the Closest Integer Value Since (n) must be a positive integer, we evaluate numbers. (n) is approximately 2.41, so we round to the next whole number (n = 3) as candidates must be integers.
The value of (n) is 3.
More Information
In combinatorial problems, understanding the difference between permutations and combinations is crucial. The permutation formula (rP_k) describes arrangements where order matters—like selecting positions for unique items—while combinations (rC_k) deals with selections where order doesn't matter.
Tips
- Confusing Permutations and Combinations: Ensure the correct formula is used based on the context (ordered vs unordered).
- Ignoring the Constraints of Integer Values: Always check if your result needs to be an integer and round correctly.
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