Number systems, permutations, combinations and probability

Number systems

• Divisibility rules: 3 and 9 (sum of digits), 4 (last two digits), 8 (last three), 11 (alternating sum difference is 0 or a multiple of 11).
• HCF × LCM = product of two numbers.
• Remainders: the remainder of a product = the product of remainders (mod n).
• Unit digits of powers repeat in cycles of 4 (2: 2, 4, 8, 6). The unit digit of 7^45: 45 mod 4 = 1 → 7.
• Number of factors of N = p^a × q^b → (a + 1)(b + 1).

Permutations and combinations

• Permutation (order matters): nPr = n! ÷ (n − r)!.
• Combination (order doesn't matter): nCr = n! ÷ (r!(n − r)!).
• Arrangements of a word with repeated letters: n! ÷ (p! q! …). "LEVEL": 5! ÷ (2! 2!) = 30.
• Circular arrangements of n people: (n − 1)!.

Example: choose a team of 3 from 5 boys and 4 girls with at least 1 girl = total − all boys = 9C3 − 5C3 = 84 − 10 = 74.

Probability

P(event) = favourable outcomes ÷ total outcomes.

• P(not A) = 1 − P(A). Often easiest for "at least one" questions.
• Independent events: P(A and B) = P(A) × P(B).
• P(A or B) = P(A) + P(B) − P(A and B).

Example: two dice. P(sum = 8) = 5/36 (2-6, 3-5, 4-4, 5-3, 6-2).

Example: P(at least one head in 3 coin tosses) = 1 − (1/2)³ = 7/8.

Data interpretation

Questions on tables, bar charts and pie charts test percentages, ratios and growth quickly. Approximate smartly (round 48.7% to 50% when options are far apart), read units and labels carefully, and calculate only what's needed.

Check yourself

• Find the unit digit of 3^102.
• In how many ways can 5 people sit around a round table?
• A bag has 4 red and 6 blue balls. P(two red when drawing two without replacement)?