How Do You Calculate Permutations for Complex Problems?

[chwala]

Homework Statement

How many even numbers less than $500$ can be formed using the digits; $[1,2,3,4,5]?$ Each digit may be used only once in any number

Relevant Equations

Permutations

Tricky questions ;

Ok in my approach;

[..., .... , ...2...] This can be filled in $3×3×1=9$ ways

[..., .... , ...4...] This can be filled in $3×3×1=9$ ways

[.... , ...4...] This can be filled in $4×1=4$ ways

[.... , ...2...] This can be filled in $4×1=4$ ways

[2] This can be filled in $1$ way

[4] This can be filled in $1$ way

Therefore number of ways is $9+9+4+4+1+1=28$ ways...

Alternatively, i was thinking of the following possibilities;

4,..., 2 The blank can be filled in $3P1×1$ ways

1,..., 2/4 The blank can be filled in $3P1×2$ ways

2,..., 4 The blank can be filled in $3P1×1$ ways

3,..., 2/4 The blank can be filled in $3P1×2$ ways

This will give us; $3P1×6=18$ ways +

[.... , ...4...] This can be filled in $4×1=4$ ways

[.... , ...2...] This can be filled in $4×1=4$ ways

[2] This can be filled in $1$ way

[4] This can be filled in $1$ way

$18+10=28$

The text gives solution only as $28.$

Your input highly appreciated. This is one of my less favorite topic...

 

[PeroK]

You can easily write down all fourteen numbers that end in $2$ and check your counting for each category.