Question

How many three digit numbers have at least one $2$ and at least one $3$?

• A. $52$
• B. $48$
• C. $54$
• D. $46$

Answer 1

The correct option is A $52$

Let U is the total number of three digit numbers

$n\left(U\right)=9\times {10}^2=900$

Let S be the number of three digit numbers not containing 2

$n\left(S\right)=8\times 9^2=648$

Let T be the number of three digit numbers not containing 3

$n\left(T\right)=8\times 9^2=648$

Number of three digit numbers not containing 2 and 3

$n(S\cap T)=7\times 8^2=448$

Now,

$n(S\cup T)=n\left(S\right)+n\left(T\right)-n(S\cap T)$

$=648+648-448=848$

Number of three digit numbers containing at least one 2 and at least one 3

$=n[U-(S\cup T\left)\right]=n\left(U\right)-n(S\cup T)$

$=900-848$

$=52$