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There's a 1 in 243 chance that he'll get them all right, and a 40 in 243 chance that he'll get 3 right.

The way this is done is we do 3^5 which is 246, and then count up the number of possibilities for each scenario. There is one chance to get all 5 questions right, and then for the wrong answers you have to consider the possibilities of getting the wrong answers wrong in multiple arrangements, such as:

0XX 0XX 0XX 0XX

0XX 0XX 0XX 0XX

0XX 0XX 0XX 0XX

X0X X0X XX0 XX0

X0X XX0 X0X XX0

And then you have to take into account that you can get different questions right, while still only having 3 questions right, such as:

0XX 0XX 0XX

0XX 0XX 0XX

0XX X0X X0X

X0X 0XX X0X

X0X X0X 0XX

When you add it all up, you get 40.

The way this is done is we do 3^5 which is 246, and then count up the number of possibilities for each scenario. There is one chance to get all 5 questions right, and then for the wrong answers you have to consider the possibilities of getting the wrong answers wrong in multiple arrangements, such as:

0XX 0XX 0XX 0XX

0XX 0XX 0XX 0XX

0XX 0XX 0XX 0XX

X0X X0X XX0 XX0

X0X XX0 X0X XX0

And then you have to take into account that you can get different questions right, while still only having 3 questions right, such as:

0XX 0XX 0XX

0XX 0XX 0XX

0XX X0X X0X

X0X 0XX X0X

X0X X0X 0XX

When you add it all up, you get 40.

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