# What is the fundamental counting principle? What is the meaning of $P_{r}^{n}$.

Last updated date: 27th Mar 2023

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Hint: Firstly write the two fundamental principles of counting then try to explain them separately using an example. In the second part write what you understand from $P_{r}^{n}.$

Complete step-by-step answer:

The fundamental principle of counting has two principles: Multiplication principle, addition principle.

i) Multiplication principle:

If an operation can be performed in ‘m’ different ways; following which a second operation can be performed in ‘n’ different ways, then the two operations in succession can be performed in $\left( m\times n \right)$ ways. This can be extended to a finite number of operations.

ii) Addition principle:

If an operation can be performed in ‘m’ different ways and another operation which is independent of the first operation can be performed in ‘n’ different ways. Then either of the two operations can be performed in ‘(m+n)’ ways. This can be extended to any finite number of mutually exclusive operations.

Meaning of $P_{r}^{n}$ :

The number of permutations of ‘n’ different things, taking ‘r’ at a time is denoted by $P_{r}^{n}.$

Note: Students should have the thorough knowledge of the theory behind counting as both the principles of counting are used in real life, while counting the number of ways to do a problem or an operation. Sometimes, they also confuse which method they should count. Students often get confused between the methods also because it is tricky to understand.

Complete step-by-step answer:

The fundamental principle of counting has two principles: Multiplication principle, addition principle.

i) Multiplication principle:

If an operation can be performed in ‘m’ different ways; following which a second operation can be performed in ‘n’ different ways, then the two operations in succession can be performed in $\left( m\times n \right)$ ways. This can be extended to a finite number of operations.

ii) Addition principle:

If an operation can be performed in ‘m’ different ways and another operation which is independent of the first operation can be performed in ‘n’ different ways. Then either of the two operations can be performed in ‘(m+n)’ ways. This can be extended to any finite number of mutually exclusive operations.

Meaning of $P_{r}^{n}$ :

The number of permutations of ‘n’ different things, taking ‘r’ at a time is denoted by $P_{r}^{n}.$

Note: Students should have the thorough knowledge of the theory behind counting as both the principles of counting are used in real life, while counting the number of ways to do a problem or an operation. Sometimes, they also confuse which method they should count. Students often get confused between the methods also because it is tricky to understand.

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