Math Problem Statement

The scheduling supervisor at Wellington Medical Center must select four nurses from 16 who are available. How many different ways can that be​ done? Question content area bottom Part 1 The nurses can be selected enter your response here different ways. ​(Type a whole​ number.)

Solution

This is a problem of combinations since the order of selecting the nurses does not matter. To calculate the number of ways to select 4 nurses from a group of 16, we use the formula for combinations:

(nr)=n!r!(nr)!\binom{n}{r} = \frac{n!}{r!(n-r)!}

Where:

  • nn is the total number of available nurses (16),
  • rr is the number of nurses to be selected (4).

Substituting the values:

(164)=16!4!(164)!=16!4!×12!\binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \times 12!}

We can cancel out the 12!12! terms from both the numerator and denominator, so:

(164)=16×15×14×134×3×2×1\binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1}

Now, perform the multiplication and division:

(164)=16×15×14×1324=4368024=1820\binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{24} = \frac{43680}{24} = 1820

Thus, there are 1820 different ways to select 4 nurses from 16.

Do you want more details or have any questions about the calculation?

Here are 5 related questions:

  1. How many ways can 5 nurses be selected from 20 nurses?
  2. What if the order of selection mattered? How would you calculate the number of arrangements (permutations)?
  3. How does the combination formula change if some selections are repeated?
  4. What is the difference between permutations and combinations?
  5. Can combinations be applied to different real-world scenarios like team formation?

Tip: When solving combination problems, remember that the order of selection doesn't matter. If order does matter, use the permutation formula instead.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Combinations

Formulas

Combination formula: C(n, r) = n! / (r!(n - r)!)

Theorems

Combination theorem

Suitable Grade Level

Grades 10-12