Good evening Candice!

For part I, that’s 4 x 2 x 5.

Will explain why later.

Part ii is 2^8 minus 1.

Will also explain later.

Good evening Mr Eric! Thank you so much. Yes, I really need help especially on part (b).

Good evening Mr Eric! Thank you so much. Yes, I really need help especially on part (b).

For part a, we view each option (number of each type of bill) as independent of one another).

How many ways are there to select the number of $10 bills to tip?

There are four ways to do so.

I could choose to tip three $10 notes...
or two $10 notes...
or one $10 note...
or none!

That's 3 + 1 = 4 ways to do so!

Regardless of my option for the $10 notes, there are two ways for the $5 bills.

I could choose to tip my $5 bill...
...or maybe not.

That's two ways to do so. Taken together with the options for the $10 bills, that's 4 x 2 = 8 possible ways.

You know what to do now for the $2 bills!

In total, there are 8 x 5 = 40 ways to do so.

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It is VERY important to note that there is no overlap in the different possible combinations. There are no two identical sums arising from different combinations in this case because a single $5 bill and four single $2 bills cannot get us $10 at all!

Had the question included five $2 bills instead, the situation gets trickier because a sum of $10, for example, can be achieved by a single $10 bill or five $2 bills.

For part b, it is similar.

There is one note of each type. For this part we just consider the different types of notes (since they did not ask for the amount to tip which can have overlaps).

For each note, we have two options. Give, or do not give.

Yes
No

Regardless of the two options for the first note, we have two options for the second note. That's 2 x 2, or 4.

Yes Yes
Yes No
No Yes
No No

Regardless of the four options for the first two notes, we have two options for the third note. That's 4 x 2, or 8.

Yes Yes Yes
Yes Yes No
Yes No Yes
Yes No No
No Yes Yes
No Yes No
No No Yes
No No No

Regardless of the eight options for the first three notes), we have two options for the fourth note. That's 8 x 2 = 16.

(You know what happens next)

This is why for eight notes, the number of combinations is 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, or simply 2^8 which is 256.

With the tipping being a must, we exclude the case of eight "no"s, so the number of possible ways is 255.