Count restricted selections and calculate exact probabilities

What counting model is used when a selected group has no order?

Unordered selections

A selection records which distinct items are chosen, not their listing order. Use combinations within each composition and add disjoint compositions. When all possible groups are equally likely, divide favourable groups by total groups using the same counting model

1. 1Ignore selection order A group and the same group written in another order are one selection \(\displaystyle \binom{5}{2}=10\)
2. 2Multiply within one composition Multiply the combination factors for the categories represented in one group \(\displaystyle \binom{4}{1}\times\binom{6}{2}=60\)
3. 3Add disjoint cases Add counts when the composition cases cannot occur in the same selection \(\displaystyle (3\times2)+(4\times2)=14\)
4. 4Use one counting model For equally likely unordered groups, build favourable and total counts using the same model before dividing \(\displaystyle \Pr(A)=\frac{\binom{4}{2}}{\binom{10}{2}}=\frac{2}{15}\)

Distinct items available

\(\displaystyle \text{red: }3,\qquad\text{blue: }3\)

Group size

\(\displaystyle r=2\)

Choose an unordered pair without replacement from the distinct items shown. How many groups contain one item of each colour?

Distinct items available

\(\displaystyle \text{red: }3,\qquad\text{blue: }3\)

Group size

\(\displaystyle r=2\)

Event

\(\displaystyle E:\text{ one of each colour}\)

Favourable groups

\(\displaystyle F=9\)

One unordered pair is selected uniformly at random without replacement from the six distinct items. Use the favourable count shown to find the probability of the displayed event

Which line correctly counts an unordered pair selected from six distinct items?