Question:
In a building, an elevator starts from the ground floor (that is, 0th floor) with 10 passengers and stops at every floor until the 15th floor which is the topmost floor. No new passengers enter the elevator on any floor from the 1st through the 15th floor. If all the passengers got off the elevator then the probability that at most one passenger got off at each floor is
In a building, an elevator starts from the ground floor (that is, 0th floor) with 10 passengers and stops at every floor until the 15th floor which is the topmost floor. No new passengers enter the elevator on any floor from the 1st through the 15th floor. If all the passengers got off the elevator then the probability that at most one passenger got off at each floor is
Show Hint
This problem is identical to distributing $n$ distinct items into $N$ distinct bins with a maximum capacity of 1.
Using the permutation notation ${}^N P_n / N^n$ makes finding the answer straightforward.
Using the permutation notation ${}^N P_n / N^n$ makes finding the answer straightforward.
Updated On: Jun 16, 2026
- $\frac{15!}{5! \times 15^{10}}$
- $\frac{10!}{15^{10}}$
- $\frac{15!}{10^{15}}$
- $\frac{15!}{5! \times 10^{15}}$
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We need to calculate the probability that 10 distinct passengers get off an elevator at 15 available floors such that no two passengers get off at the same floor.
Step 2: Key Formula or Approach:
- Each passenger has 15 choices for the floor to exit.
- The total number of ways for 10 passengers to exit is $N^n$.
- The favorable number of ways where at most one passenger exits at any floor is the permutation ${}^N P_n$.
- Probability is given by $\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$.
Step 3: Detailed Explanation:
• There are 10 passengers and 15 potential stopping floors (1st through 15th).
• Each passenger chooses a floor independently.
Thus, the total number of ways for 10 passengers to exit the elevator is:
\[ \text{Total Ways} = 15 \times 15 \times \dots \times 15 = 15^{10} \]
• For the condition "at most one passenger got off at each floor" to be satisfied, all 10 passengers must get off at 10 distinct floors.
The number of ways to choose 10 distinct floors out of 15 and assign one passenger to each is:
\[ \text{Favorable Ways} = {}^{15} P_{10} = \frac{15!}{(15 - 10)!} = \frac{15!}{5!} \]
• The required probability $P$ is:
\[ P = \frac{\text{Favorable Ways}}{\text{Total Ways}} = \frac{15!}{5! \times 15^{10}} \]
Step 4: Final Answer:
The probability is $\frac{15!}{5! \times 15^{10}}$.
We need to calculate the probability that 10 distinct passengers get off an elevator at 15 available floors such that no two passengers get off at the same floor.
Step 2: Key Formula or Approach:
- Each passenger has 15 choices for the floor to exit.
- The total number of ways for 10 passengers to exit is $N^n$.
- The favorable number of ways where at most one passenger exits at any floor is the permutation ${}^N P_n$.
- Probability is given by $\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$.
Step 3: Detailed Explanation:
• There are 10 passengers and 15 potential stopping floors (1st through 15th).
• Each passenger chooses a floor independently.
Thus, the total number of ways for 10 passengers to exit the elevator is:
\[ \text{Total Ways} = 15 \times 15 \times \dots \times 15 = 15^{10} \]
• For the condition "at most one passenger got off at each floor" to be satisfied, all 10 passengers must get off at 10 distinct floors.
The number of ways to choose 10 distinct floors out of 15 and assign one passenger to each is:
\[ \text{Favorable Ways} = {}^{15} P_{10} = \frac{15!}{(15 - 10)!} = \frac{15!}{5!} \]
• The required probability $P$ is:
\[ P = \frac{\text{Favorable Ways}}{\text{Total Ways}} = \frac{15!}{5! \times 15^{10}} \]
Step 4: Final Answer:
The probability is $\frac{15!}{5! \times 15^{10}}$.
Was this answer helpful?
0
0
Top NEST Mathematics Questions
- Let \(\theta \neq \dfrac{(2n+1)\pi}{2}\), for \(n\in\mathbb{Z}\). Consider the system of equations \[ x\cos\theta+y\sec\theta=0 \] \[ x\sin\theta+y\tan\theta=0 \] This system does not have a unique solution if and only if \(\theta\) belongs to:
- Let \(S\) be the set of all rational numbers \(r\) such that if \[ r=\frac{p}{q} \] with \(p,q\in\mathbb{Z}\), \(q\neq0\), then the two roots of the quadratic equation \[ x^{2}+2px+q^{2}=0 \] are equal. Then the number of elements in \(S\) is:
- Evaluate \[ \int_{0}^{\sqrt{\pi}} x\sin^{2}(x^{2})\,dx \]
- For \(x \in \mathbb{R}\), define \[ \langle x \rangle = x-[x] \] where \([x]\) denotes the greatest integer less than or equal to \(x\). Then for any arbitrary pair of real numbers \(x,y \in \mathbb{R}\), which of the following is always true?
- A possible solution of the system of equations \[ x^2 - 8xy + 16y^2 = 0 \] \[ (\log_{10} x)^2 + 2(\log_{10} x)(\log_{10} y) + (\log_{10} y)^2 = 4 \] is
View More Questions
Top NEST Number System Questions
- Let \(S\) be the set of all rational numbers \(r\) such that if \[ r=\frac{p}{q} \] with \(p,q\in\mathbb{Z}\), \(q\neq0\), then the two roots of the quadratic equation \[ x^{2}+2px+q^{2}=0 \] are equal. Then the number of elements in \(S\) is:
- A possible solution of the system of equations \[ x^2 - 8xy + 16y^2 = 0 \] \[ (\log_{10} x)^2 + 2(\log_{10} x)(\log_{10} y) + (\log_{10} y)^2 = 4 \] is
- Let $S = \{(x, y) \mid x, y \in \mathbb{N}, 1 \le x \le 15, 1 \le y \le 20\}$ be a set. Let $\mathcal{R}$ be the equivalence relation on $S$ defined by $(x, y) \mathcal{R} (x', y')$ if and only if $x + y = x' + y'$. Then the number of equivalence classes of $\mathcal{R}$ on $S$ is
- Let $m$ and $p$ be real numbers such that the polynomial $f(x) = x^2 + mx + p$ has two distinct negative rational roots. Then the polynomial $g(x) = x^2 - (m^2 - 2p)x + p^2$ has distinct
- Amar, Akbar, and Anthony write a test independently. The probability that Amar passes the test and Akbar fails the test is $\frac{3}{20}$. The probability that Akbar passes the test and Anthony fails the test is $\frac{1}{4}$. The probability that Amar and Anthony both pass the test is $\frac{2}{5}$. Then the probability that at least one of Amar, Akbar and Anthony fails the test is
View More Questions
Top NEST Questions
- In the 2024 Olympics, two runners, Noah Lyles and Kishane Thompson, finished the 100.00 m race in 9.784 s and 9.789 s respectively. Assuming that they move with constant speed throughout the race, the distance between the two runners when Noah Lyles touched the finishing line would be closest to:
- The Balmer series of hydrogenic spectral lines refers to an electron transitioning from \(n\geq 3\) to \(n=2\). A hydrogenic atom with atomic number \(Z=24\) undergoes a Balmer transition of the largest possible wavelength. The emitted photon has energy \(E_0\). Another hydrogenic atom with atomic number \(Z=25\) undergoes a similar Balmer transition with emitted photon energy \(E_1\). Then \(|E_1-E_0|\), in eV, is closest to:
- Three positive and two negative charges each of magnitude \(q\) are placed at five of the six vertices of a regular hexagon of side \(R\). The work done to bring a negative charge \(-q\) from infinity to the centre of the hexagon is:
- Consider a metal cylinder of heat capacity \(C\), thermal conductivity \(K\), mass \(m\) and radius \(r\). The combination that will have the dimension of time is:
- Let \(\theta \neq \dfrac{(2n+1)\pi}{2}\), for \(n\in\mathbb{Z}\). Consider the system of equations \[ x\cos\theta+y\sec\theta=0 \] \[ x\sin\theta+y\tan\theta=0 \] This system does not have a unique solution if and only if \(\theta\) belongs to:
View More Questions