Multinomial Theorem and Advanced Counting

Multinomial Theorem and Advanced Counting

This lesson covers the following key concepts:

• Multinomial theorem for (x₁ + x₂ + ... + xₖ)ⁿ
• Finding coefficient in multinomial expansion
• Rank of a word in dictionary
• Number of ways to arrange with specific positions
• Selection from identical and distinct objects
• Inclusion-Exclusion Principle
• JEE-specific advanced problems

Important Formulas

• $$(x_1 + x_2 + ... + x_k)^n = \sum \frac{n!}{r_1! r_2! ... r_k!} x_1^{r_1} x_2^{r_2} ... x_k^{r_k}$$
• $$\text{where } r_1 + r_2 + ... + r_k = n$$
• $$\text{Rank formula: use permutations of letters before}$$
• $$\text{Inclusion-Exclusion: } |A \cup B| = |A| + |B| - |A \cap B|$$
• $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|$$
• $$\text{Select from } n \text{ identical + } m \text{ distinct: } 2^m (n+1) - 1$$