Let S = {1, 2, 3, 5, 7, 10, 11}. The number o | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $S = \{1, 2, 3, 5, 7, 10, 11\}$. We classify the elements based on their remainder when divided by $3$: $R_0 = \{3\}$ (count $n_0 = 1$) $R_1 =

Vedclass · Accepted Answer

$42$

Vedclass · Answer

(A) Let $S = \{1, 2, 3, 5, 7, 10, 11\}$. We classify the elements based on their remainder when divided by $3$: $R_0 = \{3\}$ (count $n_0 = 1$) $R_1 = \{1, 7, 10\}$ (count $n_1 = 3$) $R_2 = \{2, 5, 11\}$ (count $n_2 = 3$) Let $a_n, b_n, c_n$ be the number of subsets of size $n$ with sum $\equiv 0, 1, 2 \pmod 3$ respectively. The generating function is $P(x, y) = (1+x)(1+xy)(1+xy^2)^3(1+xy)^3 = (1+x)(1+xy^2)^3(1+xy)^4$. Using the root of unity filter,the number of subsets with sum $\equiv 0 \pmod 3$ is $\frac{1}{3} [P(1, 1) + P(1, \omega) + P(1, \omega^2)]$. $P(1, 1) = 2^7 = 128$. $P(1, \omega) = (1+1)(1+\omega^2)^3(1+\omega)^4 = 2(-\omega)^3(-\omega^2)^4 = 2(-1)(\omega^8) = -2\omega^2$. $P(1, \omega^2) = (1+1)(1+\omega)^3(1+\omega^2)^4 = 2(-\omega^2)^3(-\omega)^4 = 2(-1)(\omega^4) = -2\omega$. Sum $= \frac{1}{3} [128 - 2(\omega^2 + \omega)] = \frac{1}{3} [128 - 2(-1)] = \frac{130}{3}$. Since we need nonempty subsets,we subtract the empty set (sum $0$): $\frac{130}{3} - 1 = 42.33$. Re-evaluating: The subsets are: $n_0=1, n_1=3, n_2=3$. Total subsets with sum $\equiv 0 \pmod 3$ is $43$. Excluding the empty set,the answer is $43 - 1 = 42$.

Let $S = \{1, 2, 3, 5, 7, 10, 11\}$. The number of nonempty subsets of $S$ such that the sum of all elements is a multiple of $3$ is $........$

Similar Questions

Let $S$ be the set of the first $11$ natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even}\}$ is . . . . . . .

The sum of all the elements of the set $\{\alpha \in \{1, 2, \ldots, 100\} : \operatorname{HCF}(\alpha, 24) = 1\}$ is

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails alone})$.

In a city,$20\%$ of persons read an English newspaper,$40\%$ read a Hindi newspaper,and $5\%$ read both newspapers. The percentage of people who do not read either paper is: (in $\%$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module