Question

The number of six-digit numbers that can be formed from the digits $1,2,3,4,5,6,7$ so that digits do not repeat and the terminal digits are even is

Answer 1

Terminal digits mean the first digit and the last digit. Here, it is given that the terminal digits are even.

So, The first position can be filled in $3$ ways i.e, by $2,4$ and $6$

Since the repetition is not allowed and the terminal digits need to be even, and we have used one even digit at the first position, So now we are left with '$2$‘ even digits, So, The last place can be filled in ’$2$' ways.

Now, we have left with '$5$; digits and '$4$' places. As we know that these four places can be filled with any number even or odd. But the digits cannot be repeated.
So, the possible digits for second place will be $5$
Possible digits left for third place will be $4$
Possible digits left for fourth place will be $3$
And possible digits left for the fifth place will be $2$
So, the total number of numbers that can be formed with the digits {$1,2,3,4,5,6,7$} with no digits repeated and terminal digits as even$=3\times 5\times 4\times 3\times 2\times 2$

$=720$
So, the total possible numbers will be $720$.