Combinations and Selection Problems

Combinations and Selection Problems

This lesson covers the following key concepts:

• Combination: selection where order doesn't matter
• nCr = n!/[r!(n-r)!] formula
• Properties: nCr = nC(n-r), nCr + nC(r-1) = n+1Cr
• Relation: nPr = r! × nCr
• Selection with restrictions (at least, at most)
• Distribution of identical and distinct objects
• Derangements: permutations with no fixed point

Important Formulas

• $$^nC_r = \frac{n!}{r!(n-r)!}$$
• $$^nC_r = ^nC_{n-r}$$
• $$^nC_r + ^nC_{r-1} = ^{n+1}C_r$$
• $$^nC_0 = ^nC_n = 1$$
• $$^nP_r = r! \times ^nC_r$$
• $$\text{Derangement: } D_n = n! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + ... + (-1)^n\frac{1}{n!}\right)$$