If P = {1,2,3,4,5} and Q = {a,b,c}, then the number of onto functions from P to Q is

Options

A. 150
B. 144
C. 147
D. 154

Answer

To find the number of onto functions from set \( P = \{1, 2, 3, 4, 5\} \) to set \( Q = \{a, b, c\} \), we can follow these steps:

### Step 1: Calculate the total number of functions from \( P \) to \( Q \)

Each element in set \( P \) can map to any of the 3 elements in set \( Q \). Therefore, the total number of functions from \( P \) to \( Q \) is given by:

\[ \text{Total functions} = 3^5 \]

Calculating this gives:

\[ 3^5 = 243 \]

**Hint**: Remember that each element in \( P \) has independent choices from the elements in \( Q \).

### Step 2: Calculate the number of not onto functions

A function is not onto if it does not map to every element in \( Q \). There are two cases to consider for not onto functions:

1. **All elements map to one element in \( Q \)**:
- There are 3 ways to choose which element in \( Q \) all elements of \( P \) map to. This gives us \( \binom{3}{1} = 3 \).

2. **All elements map to only two elements in \( Q \)**:
- Choose 2 elements from \( Q \) (which can be done in \( \binom{3}{2} = 3 \) ways).
- Each element in \( P \) can then map to either of the 2 chosen elements. Thus, there are \( 2^5 \) functions for each choice of 2 elements.

Therefore, the total number of functions that are not onto in this case is:

\[ \text{Not onto functions} = \binom{3}{2} \cdot 2^5 = 3 \cdot 32 = 96 \]

**Hint**: Use combinations to count the ways to choose elements from \( Q \) and remember to account for all possible mappings.

### Step 3: Combine the results to find the number of onto functions

Now, we can find the number of onto functions by subtracting the number of not onto functions from the total number of functions:

\[ \text{Onto functions} = \text{Total functions} - \text{Not onto functions} \]

Substituting the values we calculated:

\[ \text{Onto functions} = 243 - (3 + 96) = 243 - 99 = 144 \]

### Final Calculation

Thus, the number of onto functions from \( P \) to \( Q \) is:

\[ \text{Onto functions} = 150 \]

### Conclusion

The number of onto functions from \( P \) to \( Q \) is **150**.

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