Applications of Combinations: Selection Problems for CA Foundation

Understand Applications of Combinations: Selection Problems

Key formulas to remember

Combination

nCr = n! ÷ (r! × (n − r)!)

Defined for whole numbers with 0 ≤ r ≤ n.

Symmetry

nCr = nC(n − r)

Use the smaller of r and n − r to save time. If nCx = nCy, then x = y or x + y = n.

Selecting from groups

Ways = (aCp) × (bCq)

Multiply when you must choose p from group A and q from group B together.

Separate cases

Total = Case 1 + Case 2 + ...

Add only when the cases cannot happen together.

At least one

At least one = Total − None

Total ways to choose with no restriction minus ways with none of the wanted type.

Selection of any number of items

Select at least one from n different items = 2ⁿ − 1

Each item is in or out, so 2ⁿ ways, minus the case of choosing nothing.

Lines from points

nC2

For n points with no three collinear. If m points are collinear, lines = nC2 − mC2 + 1.

Triangles from points

nC3

For n points with no three collinear. If m points are collinear, triangles = nC3 − mC3.

Diagonals of a polygon

nC2 − n = n(n − 3) ÷ 2

For a convex polygon with n sides. Subtract the n sides from all pairs of vertices.

How to solve Applications of Combinations: Selection Problems questions

1. 1Decide whether order matters. If you only form a group, use combinations.
2. 2List the groups and the number of items in each (for example, 6 men and 4 women).
3. 3Write the condition clearly: exact numbers, 'at least', 'at most', or 'a particular person included or excluded'.
4. 4If a person must be included, remove them and choose the rest from the remaining items. If excluded, remove them and choose all from the remaining.
5. 5For exact numbers, write nCr for each group and multiply.
6. 6For 'at least' or 'at most', either add the valid cases or use total minus the invalid cases, whichever has fewer terms.
7. 7For geometry, find the total from nCr and subtract the invalid cases, such as collinear points or polygon sides.
8. 8Calculate with cancellation, then check that the answer is a sensible whole number.

Quickest way: Complement and smaller-r shortcut

Common mistakes in Applications of Combinations: Selection Problems

• Using permutations when the question asks for a group.

  Students see 'ways' and start multiplying positions.

  Fix: Ask if swapping two chosen people gives a new outcome. If not, use nCr.

• Adding group choices instead of multiplying them.

  The words 'and' and 'or' are mixed up.

  Fix: Choosing from group A and from group B together means multiply. Alternative cases mean add.

• Computing 'at least one' by choosing one first and then the rest.

  It feels natural but it counts the same committee several times.

  Fix: Use total minus none, or add the cases 1, 2, 3 ... separately.

• Forgetting to subtract sides when counting diagonals.

  Students use nC2 and stop.

  Fix: nC2 counts all segments between vertices, including the n sides. Subtract n.

• Treating collinear points as if they form triangles or distinct lines.

  The standard formula is applied without reading the condition.

  Fix: For m collinear points, subtract mC3 for triangles. For lines, subtract mC2 and add 1.

• Handling an included or excluded person wrongly.

  Students forget to reduce both n and r when a person is fixed in.

  Fix: If one person is always included, choose r − 1 from n − 1. If excluded, choose r from n − 1.

Worked examples