What is meant by counting principles ?

Counting principles are methods used to determine the number of possible outcomes in various situations.

What does ' AND ' typically indicate in counting problems? E.g. if you are choosing a pen and a pencil from 4 pens and 5 pencils.

AND typically indicates that values should be multiplied in counting problems. E.g. if you are choosing a pen and a pencil from 4 pens and 5 pencils, there are 4×5=20 possible ways of choosing.

What does ' OR ' typically indicate in counting problems? E.g. if you are choosing a pen or a pencil from 4 pens and 5 pencils.

OR typically indicates that values should be added in counting problems. E.g. if you are choosing a pen or a pencil from 4 pens and 5 pencils, there are 4+5=9 possible ways of choosing.

How many possible combinations are there when choosing 1 item from a list of m items AND 1 item from a list of n items ?

When choosing 1 item from a list of m items AND 1 item from a list of n items there are m × n possible combinations. This can be thought of as the fundamental counting principle.

What is the key difference between permutations and combinations ?

The key difference between permutations and combinations is that order matters in permutations but not in combinations . E.g. ACE, AEC, CAE, CEA, EAC and ECA are six separate permutations of those three letters. But those would only count as one combination of three letters.

How many possible permutations are there of n different objects?

There are n ! possible permutations of n different objects.

True or False? The word ' choose ' in a problem typically indicates a combination .

True. The word ' choose ' in a problem typically indicates a combination .

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What is meant by counting principles?

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What is meant by counting principles?
Counting principles are methods used to determine the number of possible outcomes in various situations.
What does 'AND' typically indicate in counting problems?
E.g. if you are choosing a pen and a pencil from 4 pens and 5 pencils.
AND typically indicates that values should be multiplied in counting problems.
E.g. if you are choosing a pen and a pencil from 4 pens and 5 pencils, there are 4×5=20 possible ways of choosing.
What does 'OR' typically indicate in counting problems?
E.g. if you are choosing a pen or a pencil from 4 pens and 5 pencils.
OR typically indicates that values should be added in counting problems.
E.g. if you are choosing a pen or a pencil from 4 pens and 5 pencils, there are 4+5=9 possible ways of choosing.
How many possible combinations are there when choosing 1 item from a list of m items AND 1 item from a list of n items?
When choosing 1 item from a list of m items AND 1 item from a list of n items there are m×n possible combinations.
This can be thought of as the fundamental counting principle.
True or False?
When selecting a 4-digit PIN number, the number of possibilities is greater if every digit has to be a different number.
False.
When selecting a 4-digit PIN number, the number of possibilities is smaller if every digit has to be a different number.
If each digit can be any number ('with replacement') there are $10 \times 10 \times 10 \times 10 = 10000$ possible PIN numbers.
If each digit must be different ('without replacement') there are only $10 \times 9 \times 8 \times 7 = 5040$ possible PIN numbers.
What is the key difference between permutations and combinations?
The key difference between permutations and combinations is that order matters in permutations but not in combinations.
E.g. ACE, AEC, CAE, CEA, EAC and ECA are six separate permutations of those three letters. But those would only count as one combination of three letters.
What is n! (n factorial)?
n! (n factorial) is the product of all positive integers less than or equal to n.
I.e. $n! = 1 \times 2 \times 3 \times \dots \times n$
True or False?
0! equals 0.
False.
0! equals 1.
How many possible permutations are there of n different objects?
There are n! possible permutations of n different objects.
If you arrange r out of n different objects, how many possible permutations are there?
If you arrange r out of n different objects, there are ${}^{n}P_{r} = \frac{n!}{(n - r)!}$ possible permutations.
This formula is in your exam formula booklet. However you will usually use the ${}^{n}P_{r}$ function on your GDC to calculate this, rather than using the formula.
How many possible ways are there to choose r out of n different objects?
When choosing items the order doesn't matter, so this will be a combination.
There are ${}^{n}{C_{r}} = \frac{n!}{r! (n - r)!}$ possible ways to choose r out of n different objects.
This formula is in your exam formula booklet. However you will usually use the ${}^{n}C_{r}$ function on your GDC to calculate this, rather than using the formula.
True or False?
${}^{n}{C_{r}} = {}^{n}{C_{n - r}}$
True.
${}^{n}{C_{r}} = {}^{n}{C_{n - r}}$
E.g. ${}^{6}C_{2} = {}^{6}{C_{4}} = 15$
What is the mathematical relationship between ${}^{n}{P_{r}}$ and ${}^{n}{C_{r}}$ ?
What is the mathematical relationship between ${}^{n}{P_{r}}$ and ${}^{n}{C_{r}}$ is ${}^{n}{C_{r}} = \frac{{}^{n}{P_{r}}}{r!}$ .
True or False?
The word 'choose' in a problem typically indicates a combination.
True.
The word 'choose' in a problem typically indicates a combination.

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