AlgebraHigh Yield (2 Qs in JEE)

Permutations & Combinations (P&C) Formulas

Fundamental principle of counting, nPr, nCr, circular permutations, distribution into groups, and de-arrangement.

7 Formulas•599+ Questions in Bank

Core Formulas & Mathematical Identities

{}^nP_r = \frac{n!}{(n-r)!}, \quad {}^nC_r = \frac{n!}{r!(n-r)!}, \quad {}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r

Fundamental counting formulas (Pascal's rule).

Examiner Tip: ^nC_r = ^nC_{n-r}.

x_1 + x_2 + \dots + x_r = n \implies \text{Solutions} = {}^{n + r - 1}C_{r - 1}

Stars and bars method: distributing n identical items into r distinct boxes.

Examiner Tip: For positive integral solutions (x_i ≥ 1), solutions = ^{n - 1}C_{r - 1}.

D_n = n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!} \right]

Number of permutations such that none of the n objects appear in their original natural position.

Examiner Tip: D₁ = 0, D₂ = 1, D₃ = 2, D₄ = 9, D₅ = 44.

\text{Equal Groups: } \frac{(mn)!}{(m!)^n n!}, \quad \text{Unequal Groups: } \frac{(m+n+p)!}{m! n! p!}

Formulas for partition into distinct or identical sets.

Examiner Tip: Divide by k! whenever k groups have the same size and are undistinguished.

P_{\text{circle}} = (n - 1)!, \quad P_{\text{necklace}} = \frac{(n - 1)!}{2}

Arrangements around a round table versus a necklace/garland where flip-over (clockwise = anticlockwise) makes halves identical.

Examiner Tip: If arranging n men and n women alternatively around a round table: (n - 1)! × n!.

\text{Paths from } (0, 0) \to (m, n) = \frac{(m + n)!}{m! n!} = {}^{m+n}C_m

Number of shortest monotonic paths on an m × n grid moving only rightward (R) and upward (U).

Examiner Tip: If passing through an intermediate point (p, q): Total Paths = {}^{p+q}C_p × {}^{(m-p)+(n-q)}C_{m-p}.

r^n \; (\text{Distinct items, Distinct boxes}), \quad \sum_{k=0}^r (-1)^{r-k} {}^rC_k k^n \; (\text{No empty box})

Fundamental distribution theorem with restrictions on empty containers using inclusion-exclusion.

Examiner Tip: Total number of functions from set of n elements to set of r elements is r^n; total onto functions is r! · S(n, r).

Frequently tested misconceptions in Permutations & Combinations (P&C)

•Confusing permutation (order matters) with combination (order does NOT matter).
•Circular permutations: (n - 1)! for distinct objects, (n - 1)! / 2 for garland/necklace (clockwise = counter-clockwise).