Understand Combinations

Data Science and A.I. Lecture Series

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Introduction to Combinations

A combination is a selection of items where the order does not matter.

Example: Selecting 2 players from a group of 3 players (X, Y, Z).

Possible combinations: XY, XZ, YZ.

Formula for combinations:

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Example 1: Forming Teams

How many ways can you form a team of 2 players from 4 players: A, B, C, D?

Solution:

• Total players = 4 (\(n = 4\)), Team size = 2 (\(r = 2\)).
• Using the formula:
  \[
  \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
  \]
• Combinations: AB, AC, AD, BC, BD, CD.

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Example 2: Handshakes

Twelve people meet in a room, and each shakes hands with all others. How many handshakes occur?

Solution:

• Each handshake is a combination of 2 people from 12.
• Using the formula:
  \[
  \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66
  \]
• Total handshakes = 66.

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Example 3: Drawing Cards

How many ways can 4 cards be selected from a deck of 52 cards?

Solution:

• Total cards = 52 (\(n = 52\)), Cards selected = 4 (\(r = 4\)).
• Using the formula:
  \[
  \binom{52}{4} = \frac{52!}{4!(52-4)!}
  \]
• Simplify:
  \[
  \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270,725
  \]
• Total ways = 270,725.

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Example 4: Selecting Committee Members

A committee of 3 people is to be formed from 5 men and 4 women. How many ways can this be done if the committee must include 1 man and 2 women?

Solution:

• Select 1 man from 5:
  \[
  \binom{5}{1} = 5
  \]
• Select 2 women from 4:
  \[
  \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
  \]
• Total ways:
  \[
  \binom{5}{1} \times \binom{4}{2} = 5 \times 6 = 30
  \]

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Summary

• Combinations are used when order does not matter.
• Formula:
  \[
  \binom{n}{r} = \frac{n!}{r!(n-r)!}
  \]
• Examples include forming teams, calculating handshakes, and selecting items.
• Practice makes perfect—try solving more examples to master combinations!

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Combinations

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