Adding cases

What you will be able to do

• use the addition principle for choices that are either one thing or another;
• tell when to add and when to multiply;
• split a count into separate cases and add them;
• fill the place with a rule first.

Either this or that

Kavya travels from Amli to Bhor. She can take one of $4$ buses or one of $2$ trains. How many ways does she have?

She takes a bus or a train, not both. The $4$ bus journeys and the $2$ train journeys are all different. So she has $4 + 2 = 6$ ways.

The addition principle. Suppose a task can be done in $m$ ways or in $n$ other ways, with no way counted in both. Then it can be done in $m + n$ ways.

The words "no way is counted in both" matter. Suppose $5$ students play cricket and $4$ play football. You cannot say that $9$ students play a game. Two of them might play both, and they would be counted twice.

And, or or?

The two principles answer two different questions.

Many problems need both. Kavya's journey goes on from Bhor to Chandi by one of $3$ boats. The first leg has $4 + 2 = 6$ ways. Each is followed by $3$ boats, so there are $6 \times 3 = 18$ ways in all.

Split into cases

Some counts cannot be done in one product. Split them into cases that cannot overlap. Count each case, then add.

Example. How many numbers can be made from the digits $1$, $2$, $3$, $4$ and $5$, if no digit is used twice in a number? A number may have $1$, $2$ or $3$ digits.

• One digit: $5$ numbers.
• Two digits: $5 \times 4 = 20$ numbers.
• Three digits: $5 \times 4 \times 3 = 60$ numbers.

A number has exactly one length, so no number is in two cases. The total is $5 + 20 + 60 = 85$ numbers.

Fill the place with a rule first

How many odd $3$-digit numbers have three different digits?

Start with the units place, because it has a rule. It must be odd: $1$, $3$, $5$, $7$ or $9$, so $5$ ways. Then the hundreds place cannot be $0$, and cannot repeat the units digit. That leaves $8$ ways. The tens place can use $0$, but not the two digits already used: $8$ ways.

So there are $5 \times 8 \times 8 = 320$ such numbers. If you fill the hundreds place first, you cannot tell whether the odd digits are used up. The count for the units place would then depend on the earlier choices.

A worked example

How many even $3$-digit numbers have three different digits?

The units place decides two things at once. It must be even, and if it is $0$, the hundreds place has one more choice. So split into two cases.

1. The units digit is $0$. The hundreds place has $9$ choices and the tens place $8$. That gives $9 \times 8 = 72$ numbers.
2. The units digit is $2$, $4$, $6$ or $8$. That is $4$ ways. The hundreds place cannot be $0$ or the units digit: $8$ ways. The tens place has $8$ ways. That gives $4 \times 8 \times 8 = 256$ numbers.
3. Add the cases. $72 + 256 = 328$ numbers.

Check: there are $9 \times 9 \times 8 = 648$ such numbers in all, and $648 - 320 = 328$ of them are not odd.

Practise with new numbers

Try it in Python

The program counts the numbers made from $1$ to $5$ with no digit used twice. It takes each length as a separate case, counts it by listing, and adds the cases.

Common mistakes

• Adding cases that overlap. Before you add, check that nothing can be in two cases at once.
• Adding when the choices follow each other. A shirt and then trousers is a product, not a sum.
• Filling the free places first. Fill the place with a rule first, or the count for it will change with the earlier choices.
• Forgetting the case of $0$. When $0$ is allowed in the units place but not the first, the two cases give different counts.

Remember these

• Separate cases, one OR the other: add.
• Stages one after another, first AND then: multiply.
• Split a hard count into cases that cannot overlap, then add.
• Fill a place that has a rule before the others.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A library shelf holds $7$ novels and $5$ comics. Ira borrows one book from the shelf. How many choices does she have?

2. She borrows one novel and one comic instead. How many choices does she have now?

3. How many codes of $1$ or $2$ letters can be made from A to E, with letters allowed to repeat?

4. How many $2$-digit numbers with two different digits end in $5$?

5. Of $10$ students, $6$ sing and $5$ dance. Why can you not add to say that $11$ students sing or dance?