Question

How many three digit number have at least one 2 and at least one 3.

• A. $52$
• B. $42$
• C. $44$
• D. $56$

Answer 1

The correct option is A $52$

There are three possible cases :

$1.$ Having two $2$ and one $3$ eg. $223.$

$2.$ Having two $3$ and one $2$ i.e. $332.$

Having one $2,$ one $3$ and one any other number than $2$ and $3$ eg $235.$

The option $\left(1\right)$ and $\left(2\right)$ have $3$ possible numbers each i.e. $223,232,322$ and option $\left(2\right)$ has $322,323,332.$

However for $3rd$ condition possible number is $23X,2X3,32X,3X2,X32$ and $X23,$ here $X$ is any number other than $2$ and $3.$

Where $X$ is at start we have only $7$ options but other than that we have $8$ options.

So, total no. of ways $=3+3+8\times 4+7\times 2$

$=52.$

Hence, the answer is $52.$