Permutation and Combination
Permutation and Combination
Allen Star Batch
Grade 11
Question:
Identify the correct statement(s).
Number of naughts standing at the end of $|125$ is 30
A telegraph has 10 arms each arm is capable of 9 distinct positions excluding the position of rest. The number of signals that can be transmitted is $10^9 - 1$
Number greater than 4 lacs which can be formed by using only the digits 0, 2, 2, 4, 4 and 5 is 90
In a round robin tournament, each player plays with every other player. If the number of games played is 5050 then the number of players in the tournament is 100
Step-by-Step Solution
Key Concept: Binomial expansions and algebraic identities allow systematic simplification of complex summations and coefficient extraction.
(A) From ${}^nC_1\left(\frac{x}{a}\right)\left(\frac{1}{x}\right)^5 = \frac{5}{2}$ we get $n - 6 = 0$, so $n = 6$, and $a^3 = 8$ gives $a = 2$. (B) By rewriting the double sum and simplifying using binomial identities: $S = \sum_{p=1}^4 {}^4C_p 2^{1-p} = 3^4 - 2^4 = 65$. (C) The coefficient of $x^{13}$ in $(1-x)(1-x^4)^4 = -({}^4C_3) = -4$. (D) $\sum_{r=0}^4 {}^4C_r(r-2)^2 = 16$.
Correct Answer: 2,3