Question

Counting Problem

Answer 1

\(\large \color{black}{\begin{align}
& \normalsize \text{How many 3-digit even numbers can be made using the digits}\hspace{.33em}\\~\\
& \normalsize \text{1, 2, 3, 4, 6, 7}\quad \text{ if no digit is repeated?}\hspace{.33em}\\~\\
\end{align}}\)

Answer 2

let the number be abc,

at 'c', there can be one of the 3 even numbers.
at b, there can be remaining 5 numbers
at a, there can be remaining 4 numbers.

total numbers = \(3\times 5 \times 4 \)

Answer 3

at b, there can be remaining 5 numbers

where does 5 come from ???

Answer 4

out of these 6 numbers, 1,2,3,4,6,7
one is used in the units place,
so there are 5 numbers to choose from.

Answer 5

but even numbers are three (2,4,6) not one

Answer 6

yes, thats why '3' is multiplied, not one.
but you will use only one of the 3 even numbers in the unit's place right?

Answer 7

yes

Answer 8

if we could repeat the numbers, then we would have got 6 options at both a and b position,
then it would be 6*6*3

Answer 9

but why are u starting/counting from unit's digit rather than hundred's digit

Answer 10

because there is restriction at unit's place and not at hundred's place

Answer 11

answer is 60 (given)

Answer 12

if we say there are 6 options at hundreds place, that would be incorrect,
because one of them, an even number, must be used at unit's place.

Answer 13

ohk i see ur logic

Answer 14

Alternatively, you may see that total possible 3 digit integers with distinct digits is \(6*5*4 = 120\).
However exactly half of these odd, so \(120/2 = 60\) are even.

Answer 15

You lost me... long ago. But I'm back

But you also lost me at "exactly half of these are odd"

Really? :D

Answer 16

we can say exactly half of these are odd,
only because half of the given digits are odd!
otherwise, that statement won't hold true.

Answer 17

Haha exactly ^

Answer 18

Sorry... I keep missing that there's no 5 in the set.

Dang misleading questions...

Answer 19

Though clever...

Answer 20

the given set has some sort of symmetry with respect to even/odd which we can use to our advantage :)