Academia Aeternum
तमसो मा ज्योतिर्गमय
Count Every Arrangement — 42 P&C Problems with Full Reasoning
5 exercise files · 42 total questions
\(^nP_r = \dfrac{n!}{(n-r)!}\)\(^nC_r = \dfrac{n!}{r!\,(n-r)!} = {}^nC_{n-r}\)\(\text{Circular} = (n-1)!\)\(^nC_r + {}^nC_{r-1} = {}^{n+1}C_r \text{ (Pascal)}\)\(\text{At-least-one} = \text{Total} - \text{(none selected)}\)Step 1 — Ask 'Does ORDER matter?' Yes→Permutation, No→Combination. Step 2 — Restricted: fix constrained objects first, arrange rest. Step 3 — 'Always together': treat group as 1 unit, arrange, then internally arrange. Step 4 — 'Never together': Total − (arrangements where they ARE together). Step 5 — Rank of word: count letters alphabetically less than each position.
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