Twenty persons arrive in a town having 3 hote | vedclass

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

Question

(D) Let $n_x, n_y, n_z$ be the number of persons in hotels $x, y, z$ respectively. We have $n_x + n_y + n_z = 20$. We want to find the number of ways

Vedclass · Accepted Answer

$\frac{{3^{20} - 13 \cdot 2^{20} + 43}}{{3^{20}}}$

Vedclass · Answer

(D) Let $n_x, n_y, n_z$ be the number of persons in hotels $x, y, z$ respectively. We have $n_x + n_y + n_z = 20$. We want to find the number of ways such that $n_x \geq 2, n_y \geq 1, n_z \geq 1$.Total ways = $3^{20}$.Let $S$ be the total set of outcomes. Let $A$ be the condition $n_x $n(B) = 2^{20}$ (all in $x$ or $z$),$n(C) = 2^{20}$ (all in $x$ or $y$).$n(A) = n(n_x=0) + n(n_x=1) = 2^{20} + 20 \cdot 2^{19}$.Using the Principle of Inclusion-Exclusion for the complement set (where the condition is $NOT$ met),the number of favorable cases is $3^{20} - [n(A) + n(B) + n(C) - (n(A \cap B) + n(B \cap C) + n(C \cap A)) + n(A \cap B \cap C)]$.After calculating the intersections and applying the formula,the number of favorable cases is $3^{20} - 13 \cdot 2^{20} + 43$.Thus,the probability is $\frac{3^{20} - 13 \cdot 2^{20} + 43}{3^{20}}$.

Twenty persons arrive in a town having $3$ hotels $x, y$ and $z$. If each person randomly chooses one of these hotels,then what is the probability that at least $2$ of them go to hotel $x$,at least $1$ to hotel $y$,and at least $1$ to hotel $z$? (Each hotel has a capacity for more than $20$ guests.)

Similar Questions

Let $A$ denote the set of all $2$-digit numbers in base $10$ that are equal to four times the sum of the factorials of their digits. The sum of the numbers in $A$ is

Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers satisfying the condition $x^2 - y^2 = 12345678$. Then,

The number of integers from $1$ to $1000$ which are divisible by $2$ or $3$ is

$A$ and $B$ are two subsets of set $S = \{1, 2, 3, 4\}$ such that $A \cup B = S$. Then,the number of ordered pairs $(A, B)$ is:

Out of $500$ car owners investigated,$400$ owned car $A$ and $200$ owned car $B$,and $50$ owned both car $A$ and car $B$. Is this data correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module