Permutation and Combination
Permutation and Combination
DAILY_CHALLENGE
Grade 11
Question:
Number of ways in which the letters of the word 'B U L B U L' can be arranged in a line in any order is also equal to the
number of ways in which 2 alike Apples and 4 alike Mangoes can be distributed in 3 children so that each child receives any number of fruits.
Number of ways in which 6 different books can be tied up into 3 bundles, if each bundle is to have equal number of books
coefficient of $x^3y^2z^2$ in the expansion of $(x + y + z)^6$
number of ways in which 6 different prizes can be distributed equally in three children.
Step-by-Step Solution
Key Concept: Irrational conjugate pairs raised to integer powers produce integer parts and fractional parts with special relationships.
Given $(5+\sqrt{2})^n + (5-\sqrt{2})^n = 2k$ where $k \in \mathbb{I}$, we note that $\{x\} + \{x\} = N$ with $0 < \{x\} < 1$ implies $\{x\} + N = 1$. Therefore $x^2 + \{x\} = x - x(1-\{x\}) = x(1-\{x\})$, and $N = (5+2\sqrt{6})^n(5-2\sqrt{6})^n - 1$.
Correct Answer: 1,3,4