Question

Find the number of all onto functions from the set {1,2,3,....,n} to itself​

Answer 1

Consider a set {1,2,3}

Since f is onto, all elements of a set will have unique pre-images.

Element Number of possible pairings

1 - 3

2 - 2

3 - 1

Total number of one-one function = 3 × 2 × 1 = 6

Now consider a set {1,2,3………n}

Since f is onto, all elements of a set will have unique pre-images.

Element Number of possible pairings

1 - 3

2 - 2

3 - 1

. - .

. - .

n-1 - 2

n - 1

Total number of one-one function = n × (n – 1) × (n – 2) × …….× 2 × 1 = n!

Answer 2

Consider a set {1,2,3}

Since f is onto, all elements of a set will have unique pre-images.

Element

Number of possible pairings

1

3

2

2

3

1

Total number of one-one function = 3 × 2 × 1 = 6

Now consider a set {1,2,3………n}

Since f is onto, all elements of a set will have unique pre-images.

Element

Number of possible pairings

1

3

2

2

3

1

.

.

.

.

n – 1

2

n

1

Total number of one-one function = n × (n – 1) × (n – 2) × …….× 2 × 1 = n!