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junior college 2 | H2 Maths

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Hi, pls help on this question, thank you so much!

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.

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.

-------------------------------------------------

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.

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.

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Does the diagram resemble something which you have learnt during secondary level Maths?

For every case, we branch out to two further outcomes. This will be why the number of variations will be 2^8 - and there are no identical scenarios also so we do not have to worry.

We remove the one case where all eight notes are not tipped.

Thank you so much for your awesome explanation. The diagram helps alot! I can understand instantly just by looking at your drafted diagram :)

Hope you have a great day ahead.