Question

A letter is selected from each of the words '$MATHEMATICS$ ' and '$STATISTICS$'. The probability that the same letters are selected is

• A. $\frac{7}{55}$
• B. $\frac{9}{55}$
• C. $\frac{12}{55}$
• D. None of these
  

Answer 1

The correct option is A $\frac{7}{55}$

We are interested in selecting a letter from the word ${}^{\prime }MATHEMATIC{S}^{\prime }$ and a letter from the word ${}^{\prime }STATISTIC{S}^{\prime }$.
Number of letters in the two words are $11$ and $10$.
A letter from the two words can be taken in ${}^{11}{C}_1{\times }^{10}{C}_1$ ways.
But the letters should be in both the words. The letters common for both are $A,T,I,C$ and$S$
These are $2A^{\prime }s$ in ${}^{\prime }MATHEMATIC{S}^{\prime }$ and one $A$ in ${}^{\prime }STATISTIC{S}^{\prime }$.
So, $A$ can be selected in $2^2{C}_1{\times }^1{C}_1$ ways.
Similarly, $T$ can be selected in ${}^2{C}_1{\times }^3{C}_1$ ways.
$I$ can be selected in ${}^1{C}_1{\times }^1{C}_1$ and $\sin {}^1{C}_1{\times }^3{C}_1$ ways.
Either of the letter is selected. So favourable events are
${}^2{C}_1{\times }^1{C}_1{+}^2{C}_1{\times }^3{C}_1{+}^1{C}_1{\times }^2{C}_1{+}^1{C}_1{\times }^3{C}_1{+}^1{C}_1{\times }^1{C}_1$
$\therefore$ $P$(selecting the same letter from both the words)
$=\frac{{}^2{C}_1{\times }^1{C}_1{+}^2{C}_1{\times }^3{C}_1{+}^1{C}_1{\times }^2{C}_1{+}^1{C}_1{\times }^3{C}_1{+}^1{C}_1{\times }^1{C}_1}{{}^{11}{C}_1{\times }^{10}{C}_1}$
$=\frac{14}{110}=\frac{7}{55}$