Combinations — A/L Combined Mathematics

What must I remember about Combinations?

• Definition of Combination: A selection of objects without regard to order (order of selection DOES NOT matter).
• Combinations formula: Number of combinations of n distinct objects chosen r at a time: nCr = n! / [r! * (n - r)!] = nPr / r!.
• Fundamental relationship between permutations and combinations: nPr = nCr * r! (selecting r items, then arranging them in r! orders).
• Properties of combinations: nCr = nC(n-r); nC0 = nCn = 1; nC1 = n; Pascal's identity: nCr + nC(r-1) = (n+1)Cr.
• Team / Committee formation: Selecting r members from group of n under specific category quotas: choose from subgroup A, then multiply by selections from subgroup B.

Test yourself (1 minute)

1. Equality condition for combinations

   If nCx = nCy, then either x = y or x + y = n.

2. Division into groups

   Number of ways to divide (m + n) distinct items into two groups of sizes m and n (m ≠ n) is (m+n)! / (m! n!); if group sizes are equal (m = n), divide by 2! to prevent overcounting.

3. Geometric points configurations

   From n points with no 3 collinear: Number of straight lines = nC2; Number of triangles = nC3; Number of diagonals of n-sided polygon = nC2 - n = n(n - 3)/2.

Practise Combinations questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 23

What should I study first?

What comes next?

Next in the syllabus

Where does Combinations fit in the syllabus?

Year 2 · Term 1