Math Problem Statement

How many unordered sets are there of four items chosen from eight? sets

Solution

To determine the number of unordered sets of four items chosen from a total of eight, we use the concept of combinations. For an unordered set (where the order of selection doesn't matter), the formula for combinations is:

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

where:

  • n=8n = 8 (the total number of items),
  • r=4r = 4 (the number of items to choose).

Plugging in the values:

(84)=8!4!(84)!=8!4!4!\binom{8}{4} = \frac{8!}{4!(8 - 4)!} = \frac{8!}{4! \cdot 4!}

Now, calculate each part:

  1. 8!=8×7×6×5×4!8! = 8 \times 7 \times 6 \times 5 \times 4!, but since it divides by 4!4!, we can simplify this to 8×7×6×58 \times 7 \times 6 \times 5.

  2. Calculating the above:

    8×7=568 \times 7 = 56 56×6=33656 \times 6 = 336 336×5=1680336 \times 5 = 1680

  3. Now divide by 4!=244! = 24:

    168024=70\frac{1680}{24} = 70

So, there are 70 unordered sets of four items chosen from eight.

Would you like further details or have any questions?

Related Questions

  1. How would the answer change if we chose 5 items from 8 instead of 4?
  2. What if we chose ordered sets instead of unordered sets?
  3. How many unordered sets are there of three items chosen from eight?
  4. What is the total number of combinations if we choose two items from eight?
  5. How does this calculation change for a scenario with replacement?

Tip:

When solving combination problems, always simplify factorial expressions as much as possible before multiplying to make calculations easier.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Combinations

Formulas

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

Theorems

Basic counting principle in combinatorics

Suitable Grade Level

Grades 10-12