Question

A letter is taken out at random from ‘$ASSISTANT$’ and another is taken out from ‘$STATISTICS$’. The probability that they are the same letters, is

• A. $\frac{1}{45}$
• B. $\frac{13}{90}$
• C. $\frac{19}{90}$
• D. None of these

Answer 1

The correct option is C

$\frac{19}{90}$

The explanation for the correct option:

From the words “$ASSISTANT$” and “$STATISTICS$”, the same letters are $A,I,S,T$

Therefore,

The probability of choosing $A$

$\begin{array}{rcl}& =& \frac{{}^{2}C_1}{{}^{9}C_1}.\frac{{}^{1}C_1}{{}^{10}C_1}\\ & =& \frac{1}{45}\end{array}$

The probability of choosing $I$

$\begin{array}{rcl}& =& \frac{{}^{1}C_1}{{}^{9}C_1}.\frac{{}^{2}C_1}{{}^{10}C_1}\\ & =& \frac{1}{45}\end{array}$

The probability of choosing $S$

$\begin{array}{rcl}& =& \frac{{}^{3}C_1}{{}^{9}C_1}.\frac{{}^{3}C_1}{{}^{10}C_1}\\ & =& \frac{1}{10}\end{array}$

The probability of choosing $T$

$\begin{array}{rcl}& =& \frac{{}^{2}C_1}{{}^{9}C_1}.\frac{{}^{3}C_1}{{}^{10}C_1}\\ & =& \frac{1}{15}\end{array}$

The total probability is

$\begin{array}{rcl}& =& \frac{1}{45}+\frac{1}{45}+\frac{1}{10}+\frac{1}{15}\\ & =& \frac{19}{90}\end{array}$

Therefore, correct answer is option (C).