Question

A letter is selected from each of the words '$MATHEMATICS$' and '$STATISTICS$'. The probability that the same letters are selected is $k/55$. Find the value of $k$.

Answer 1

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