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 will use the principle of inclusion-exclusion.

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

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

\[ \text{Total functions} = 3^5 = 243 \]

**Hint:** Remember that for each element in \( P \), there are multiple choices in \( Q \).

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

A function is not onto if at least one element in \( Q \) is not mapped to by any element in \( P \). We will consider cases where one or more elements of \( Q \) are excluded.

**Case 1:** Functions that map to only 1 element in \( Q \)

If all elements of \( P \) map to a single element in \( Q \), there are 3 such functions (all to \( a \), all to \( b \), or all to \( c \)).

\[ \text{Not onto (1 element)} = 3 \]

**Case 2:** Functions that map to exactly 2 elements in \( Q \)

We can choose 2 elements from \( Q \) in \( \binom{3}{2} = 3 \) ways. For each of these pairs, each of the 5 elements in \( P \) can map to either of the 2 chosen elements. Thus, the number of functions that map to exactly 2 elements is:

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

**Hint:** Use combinations to select which elements of \( Q \) are included in the mapping.

### Step 3: Apply the principle of inclusion-exclusion

Now, we can find the total number of functions that are not onto by adding the two cases:

\[ \text{Total not onto} = \text{Not onto (1 element)} + \text{Not onto (2 elements)} = 3 + 96 = 99 \]

### Step 4: Calculate the number of onto functions

The number of onto functions is the total number of functions minus the number of functions that are not onto:

\[ \text{Onto functions} = \text{Total functions} - \text{Total not onto} = 243 - 99 = 144 \]

### Final Answer

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

\[ \boxed{144} \]