Question

If \[s=\left\{ 2,3,4,5,7,9 \right\}\] , then the number of different three digit numbers (with all distinct digits) lease than \[400\] that can be formed from s, is

Answer 1

Hint: To find the total number of possibilities we need to first need to place the number of possibilities that a number can take place in a three digit number system. The first place can have a number less than \[4\] as the number of three digits should be less than \[400\] , similarly other combinations are to be calculated by placing each digit in the rest of the two places of the three digit number.

Complete step-by-step answer:
Let us check for the first place of the three digit number, the number of possibilities that the first place can hold from the numbers given in s is \[ 2,3\] as the numbers less than \[400\] is from \[0\] to \[399\] . Hence, the first digit has to be less than \[4\] and the number of such possibilities is \[2\] .
Now moving on to the second position of the number the second position can hold at least \[4\] numbers and one extra depending on the first digit and as the number can’t be repeated therefore, the number of ways digits can be put in the middle space is \[5\] .
Similarly, for the last place the total number of possibilities for the middle space is \[5\] then subtracting a single number will give the possibility of the last space as \[4\] .
Now as we have all the possibilities of the numbers that can be put in the three digit number we can say that the total number of possibility of the three digit number from the series of number \[s=\left\{ 2,3,4,5,7,9 \right\}\] is given as the product of
 \[2\times 5\times 4=40\] .
Therefore, the number of different three digit numbers (with all distinct digits) less than \[400\] that can be formed from s, is \[40\] .
So, the correct answer is “40”.

Note: Students may get wrong results if they take the numbers in repeating form, remember the numbers that are given in \[s=\left\{ 2,3,4,5,7,9 \right\}\] are the only digits we can choose and not repeat anyone of them.