Math Problem Statement

^nC1

Solution

The expression nC1^nC_1 represents a combination, specifically the number of ways to choose 1 item from a set of nn distinct items. The formula for combinations is given by:

nCr=n!r!(nr)!^nC_r = \frac{n!}{r!(n-r)!}

For nC1^nC_1:

nC1=n!1!(n1)!^nC_1 = \frac{n!}{1!(n-1)!}

Since 1!1! is 1 and (n1)!(n-1)! cancels out with part of n!n!, we have:

nC1=n×(n1)!1×(n1)!=n^nC_1 = \frac{n \times (n-1)!}{1 \times (n-1)!} = n

So, nC1=n^nC_1 = n.

This means there are nn ways to choose 1 item from a set of nn items.

Would you like more details on combinations or how they are used in different scenarios?

Here are 5 related questions:

  1. What is the formula for nCr^nC_r and how is it derived?
  2. How do combinations differ from permutations?
  3. How would you calculate nC2^nC_2 for a specific value of nn?
  4. Can you explain the real-world application of combinations in probability?
  5. What happens to nCr^nC_r when r=nr = n?

Tip: Remember that combinations count selections where the order doesn't matter, unlike permutations, where order is important.

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Math Problem Analysis

Mathematical Concepts

Combinations
Factorials

Formulas

Combination formula nCr = n! / (r!(n-r)!)

Theorems

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Suitable Grade Level

Grades 9-12