Dabour Hub EGEgyptian Curriculum 2026/2027
Recall: counting principle, permutations and combinations

Recall: counting principle, permutations and combinations

Point 1 of 7 — Lesson 1: Counting Principle, Permutations (nPr) and Combinations (nCr)Question bank: 6 questions (6 with pictures)

01 Recall: counting principle, permutations and combinations

Counting principle

  • Multiplication rule (AND): if one task can be done in m ways and, after it, a second task in n ways, the two tasks together can be done in m×n ways.
  • Addition rule (OR): if two tasks cannot happen together and they can be done in m and n ways, then one of them can be done in m+n ways.
  • Factorial: n!=n(n−1)(n−2)⋯3×2×1, with 0!=1.

Permutations (nPr)

  • Permutation: an arrangement of r objects chosen from n different objects, where the order matters. nPr=n!(n−r)!=n(n−1)(n−2)⋯(n−r+1)
  • nPn=n! and nP1=n.
  • Arranging n different objects around a circle (no fixed reference point) gives (n−1)! ways.

Combinations (nCr)

  • Combination: a selection of r objects out of n different objects, where the order does not matter. nCr=n!r! (n−r)!=nPrr!
  • nCr=nCn−r, and nC0=nCn=1,  nC1=n.
  • If nCx=nCy, then either x=y or x+y=n.
  • Pascal's rule: nCr+nCr−1=n+1Cr.
  • Ratio of consecutive combinations: nCrnCr−1=n−r+1r.
Point of comparison Permutations nPr Combinations nCr
Order Matters Does not matter
Formula n!(n−r)! n!r!(n−r)!
Typical use Forming numbers, passwords, rankings Choosing committees, cards, subsets
Size nPr≥nCr nCr≤nPr
✍️ Worked examples

Example 1:

  • How many different 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 without repetition?
  • In how many ways can a committee of 3 be chosen from 7 people?

Solution:

  • Numbers: the order matters, so 5P3=5×4×3=60.
  • Committee: the order does not matter, so 7C3=7×6×53×2×1=35.

📝 Question bank for this point

6 questions linked to this point only, 6 of them with pictures.