Question

The first 12 letters of the English alphabets are written down at random. The probability that there are 4 letters between A & B is:
A. $ \dfrac{7}{{33}} $
B. $ \dfrac{{12}}{{33}} $
C. $ \dfrac{{14}}{{33}} $
D. $ \dfrac{7}{{66}} $

Answer 1

Hint: To find the required probability, we use the concept of permutation and combination. We will require the expression of combination which is given as where is the total number of things and is the things that need to be chosen. Since we have two letters already arranged, we will require four more letters to be arranged in between. We will use the concept of arrangements of objects as per the required situation. We will also use the concept of probability according to which it can be found out by the ratio of the required number of ways to the total number of ways.

Complete step-by-step answer:
Since we have 12 letters, therefore, these 12 letters can be arranged as expressed:
 $ {\rm{Total no}}{\rm{. of ways}} = \left| \!{\underline {\,
  {12} \,}} \right. $
 In the question, it is given that 4 letters are to be arranged between A and B. This can be expressed as:
A __ __ __ __ B
So, these 4 letters can be arranged in $ \left| \!{\underline {\,
  4 \,}} \right. $ ways and also letters A and B arranged in $ \left| \!{\underline {\,
  2 \,}} \right. $ ways. Since, A and B two letters are already selected, therefore we are left with 10 letters. Out of these 10 letters we need to select 4 letters which can be expressed as:
  $ {}^{10}{C_4}\,\left| \!{\underline {\,
  4 \,}} \right. \;\left| \!{\underline {\,
  2 \,}} \right. $
Now we have 6 letters remaining along with 1 formed above. The arrangement of these 7 letters can be expressed as:
 $ {\rm{Required arrangement}} = \left| \!{\underline {\,
  7 \,}} \right. \;\left( {{}^{10}{C_4}\,\left| \!{\underline {\,
  4 \,}} \right. \;\left| \!{\underline {\,
  2 \,}} \right. } \right) $
 The probability is the ratio of required number of ways of arrangement to the total number of ways of arrangement. The expression for probability can be given as:
 $ {\rm{Probability}} = \dfrac{{\left| \!{\underline {\,
  7 \,}} \right. \left( {{}^{10}{C_4}\,\left| \!{\underline {\,
  4 \,}} \right. \;\left| \!{\underline {\,
  2 \,}} \right. } \right)}}{{\left| \!{\underline {\,
  {12} \,}} \right. }} $
We know that combination of arrangements can be expressed as:
 $ {}^n{C_r} = \dfrac{{\left| \!{\underline {\,
  n \,}} \right. }}{{\left| \!{\underline {\,
  {n - r} \,}} \right. \,\left| \!{\underline {\,
  r \,}} \right. }} $
We will use this expression in the expansion of $ {}^{10}{C_4} $ in the expression of probability.
 $ \begin{array}{l}
{\rm{Probability}} = \dfrac{{\left| \!{\underline {\,
  7 \,}} \right. \left( {\dfrac{{\left| \!{\underline {\,
  {10} \,}} \right. }}{{\left| \!{\underline {\,
  6 \,}} \right. \,\left| \!{\underline {\,
  4 \,}} \right. }} \times \,\left| \!{\underline {\,
  4 \,}} \right. \;\left| \!{\underline {\,
  2 \,}} \right. } \right)}}{{\left| \!{\underline {\,
  {12} \,}} \right. }}\\
{\rm{Probability}} = \dfrac{{\left| \!{\underline {\,
  7 \,}} \right. \,\left| \!{\underline {\,
  {10} \,}} \right. \,\left| \!{\underline {\,
  4 \,}} \right. \;\left| \!{\underline {\,
  2 \,}} \right. }}{{\left| \!{\underline {\,
  {12} \,}} \right. \,\left| \!{\underline {\,
  6 \,}} \right. \,\left| \!{\underline {\,
  4 \,}} \right. }}\\
{\rm{Probability}} = \dfrac{{7 \times \left| \!{\underline {\,
  6 \,}} \right. \times \,\left| \!{\underline {\,
  {10} \,}} \right. \times \,\left| \!{\underline {\,
  4 \,}} \right. \times \;\left| \!{\underline {\,
  2 \,}} \right. }}{{12 \times 11 \times \left| \!{\underline {\,
  {10} \,}} \right. \, \times \left| \!{\underline {\,
  6 \,}} \right. \times \,\left| \!{\underline {\,
  4 \,}} \right. }}\\
{\rm{Probability}} = \dfrac{{7 \times \;2}}{{12 \times 11\,}}\\
{\rm{Probability}} = \dfrac{7}{{66\,}}
\end{array} $
Hence, the probability that there are 4 letters between A & B is $ \dfrac{7}{{66}} $ .
So, the correct answer is “Option D”.

Note: It is essential to identify whether we are concerned with only arrangements, or we are also considering the order of arrangements. If we are dealing with an only selection of objects, we will use the concept of combination, but if we have to select the objects and along with that we have to find the order of arrangements, then we will use the concept of permutation.