Question

The number of four digit numbers formed by using the digits 0, 2, 4, 5 and which are not divisible by 5, is:
A.10
B. 8
C. 6
D. 4

Answer 1

Hint: In the question we have been given that the number must not be divisible by 5 that is the number must not end with the digit 0 and 5. So that it violates the divisibility rule of 5. Also the number of distinct digits given to us are four (0, 2, 4, 5) to be used to make four digit numbers.

Complete step by step solution:
The above question is an example of restricted permutation. That is permutations of n different things taken r at a time, where x particular things are always excluded which in this case are the numbers divisible by 5. Therefore we have four places to fill to make a four digit number
_ _ _ _
But we have to make sure that the thousandth place is not filled by the digit 0 and the left three places can be filled by any of the left three digits since one digit out of 2, 4 or 5 leaving only three digits
Therefore total number of four digits numbers will be given by
 $
\Rightarrow {^4}{P_3} - \left| \!{\underline {\,
  3 \,}} \right. = \dfrac{{\left| \!{\underline {\,
  4 \,}} \right. }}{{\left| \!{\underline {\,
  {(4 - 3)} \,}} \right. }} - 3 \times 2 \times 1 \\
   = \dfrac{{\left| \!{\underline {\,
  4 \,}} \right. }}{{\left| \!{\underline {\,
  1 \,}} \right. }} - 6 = 4 \times 3 \times 2 \times 1 - 6 = 24 - 6 = 18 \\
 $
Where $ \left| \!{\underline {\,
  3 \,}} \right. $ gives the different arrangements of the three digits in the left three places.
Now the restrictions come into play the last place should not have digit 5 and can have any of the other leftover digits except zero leaving only two digits to rearrange therefore the arrangements will be
 $ \left| \!{\underline {\,
  3 \,}} \right. - \left| \!{\underline {\,
  2 \,}} \right. = 4 $
Also the last place should not have the digit 0 and can have any other three digits that is given by $ \left| \!{\underline {\,
  3 \,}} \right. $ . The total restricted arrangements will be
6+4=10
So the number of four digits numbers formed from the digits 0, 2, 4, 5 will be
 $ 18 - 10 = 8 $
So, the correct answer is “Option B”.

Note: We need to make sure that in such cases of restricted permutations we have to subtract the restricted arrangements from the arrangements that would have been possible if there would have been no restrictions, in this case the number divisible with 5 are subtracted from the total number of four digit formed from the given digits.