Permutations & combinations

Story hook

"In how many ways can the letters of the word 'HEALTHCARE' be arranged?"

You count letters: H, E, A, L, T, H, C, A, R, E → 10 letters. But some are repeated: H twice, E twice, A twice.

Wrong answer: 10! = 36,28,800.

Right answer: 10! / (2! × 2! × 2!) = 36,28,800 / 8 = 4,53,600.

A 30-second formula. Without it, candidates spend 5 minutes listing and miscounting.

Permutations + Combinations + Probability = the "counting trinity". 4-6 questions per CSAT paper. The hardest part isn't the formulas — it's deciding which one to use: order matters (permutation) or doesn't matter (combination)?

After this unit, you'll decide in 5 seconds which formula applies, and solve in 30.

Why this matters for UPSC

For CSAT (Paper II, qualifying 33%):

• 4-6 questions per paper are P&C/Probability. Often appear as "arrangements", "selections", "card/dice/coin" problems.
• P&C intuition crosses into Reasoning (the other CSAT paper) — arrangements of people in a row, around a table.
• Real life: Probability is the bedrock of risk thinking in policy (epidemiology, insurance, agriculture forecasting).

Getting all 4-6 P&C questions right = +8-12 marks in CSAT. 1 hour of focused practice is enough to nail the patterns.