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14. probability — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS14. probability1 Mark
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 ${}^n{C_3}.\frac{{(n - 3)\,!}}{{n\,!}} = \frac{1}{6}$.

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