Download App

STD 11 - 14. probability — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 14. probability4 Marks
MCQ
There are $n$ different objects $1, 2, 3,......n$ distributed at random in $n$ places marked $1, 2, 3, ......n$. The probability that at least three of the objects occupy places corresponding to their number is
  • ✓
    $\frac{1}{6}$
  • B
    $\frac{5}{6}$
  • C
    $\frac{1}{3}$
  • D
    None of these
✓

Answer

Correct option: A.
$\frac{1}{6}$
a
(a) Let ${E_i}$ denote the event that the ${i^{th}}$ object goes to the ${i^{th}}$ place,

we have $P({E_i}) = \frac{{(n - 1)\,!}}{{n\,!}} = \frac{1}{n},\forall \,\,i$

and $P({E_1} \cap {E_j} \cap {E_l}) = \frac{{(n - 3)\,\,!}}{{n\,\,!}}$ for $i < j < k$

Since we can choose $3$ places out of $n$ in ${}^n{C_3}$ ways.

The probability of the required event is ${}^n{C_3}.\frac{{(n - 3)\,!}}{{n\,!}} = \frac{1}{6}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

$\int_{}^{} {\frac{{1 + {{\cos }^2}x}}{{{{\sin }^2}x}}dx} = $
→
Let the functions $:(-1,1) \rightarrow R$ and $g:(-1,1) \rightarrow(-1,1)$ be defined by $f(x)=|2 x-1|+|2 x+1|$ and $g(x)=x-[x]$,

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ:(-1,1) \rightarrow R$ be the composite function defined by $(f \circ g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is $NOT$ differentiable. Then the value of $c+d$ is. . . . .

→
If the coefficients of $x^4$, $x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:
→
$\int_{ - \pi /2}^{\pi /2} {\sqrt {\frac{1}{2}(1 - \cos 2x)} } \,dx = $
→
Let $A =\left(\begin{array}{ll}2 & -2 \\ 1 & -1\end{array}\right)$ and $B =\left(\begin{array}{ll}-1 & 2 \\ -1 & 2\end{array}\right)$. Then the number of elements in the set $\left\{( n , m ): n , m \in\{1,2, \ldots . .10\}\right.$ and $\left.nA ^{ n }+ mB ^{ m }= I \right\}$ is
→
If $r = {[2\phi + {\cos ^2}(2\phi + \pi /4)]^{1/2}}$ then what is the value of the derivative of $dr/d\phi  $ at $\phi = \pi /4$
→
Eccentricity of the rectangular hyperbola $\int_0^1 {{e^x}\left( {\frac{1}{x} - \frac{1}{{{x^3}}}} \right)} \;dx$ is
→
Let $g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime}(x)>0$ for all $\mathrm{x} \in(0,3)$. If $\mathrm{g}$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8 \alpha$ is
→
If $A_n =$ $\int\limits_0^{\pi /2} \frac{{\sin \,\,(2\,n\,\, - \,\,1)\,\,x}}{{\sin \,\,x}} d x$ ; $B_n =$ $\int\limits_0^{\pi /2}\, {\left( {\frac{{\sin \,\,n\,x}}{{\sin \,\,x}}} \right)^2} d x$ ; for $n \in N$ , then :
→
Let $a_1, a_2, \ldots . . a_{10}$ be 10 observations such that $\sum_{ k =1}^{10} a _{ k }=50$ and Image. Then the standard deviation of $a_1, a_2, \ldots, a_{10}$ is equal to :
→