A child paints the six faces of a cube with six different co

Number of ways to paint a cube with 6 distinct colours:
Put any one colour of any one side in 1 way. (Say Red). Rest the cube on this side. It is the bottom.
Now, there are 5 ways to place a colour opposite red (top) which can be done in 5 ways.
All other 4 sides are identical so place any one remaining colour on any 1 side in 1 way. (say pink is placed here)
Now leftover 3 sides are distinct (one side to left of pink, one to the right and one opposite to pink).
So 3 colours can be placed in 3! ways.
Total = 5 * 3!

Number of ways to put Red, pink and blue on a common corner:
Put Red on any side in 1 way and place it at the bottom. The number of ways of placing a colour opposite to red (top) is 3 (we cannot place pink or blue there).
Now we have 4 identical sides left. On any one, place pink in 1 way. Now we have 3 distinct sides. Blue must be placed on one of the 2 adjoining pink. This can be done in 2 ways. Now we have 2 sides and 2 colours left which can be placed in 2 ways.
Total = 3 * 2 * 2

Required Probability = 2/5

Not sure why the options don't have 2/5. Can't see any error in my reasoning either.

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