If A={1,2,3}, B={1,3,5,7,9}, the ratio of number of one-one functions to the number of strictly monotonic functions is ………. .

Answer

To solve the problem of finding the ratio of the number of one-one functions to the number of strictly monotonic functions from set A to set B, we will follow these steps:

### Step 1: Identify the Sets
Given:
- Set A = {1, 2, 3}
- Set B = {1, 3, 5, 7, 9}

### Step 2: Calculate the Number of One-One Functions
A one-one function (or injective function) maps distinct elements of set A to distinct elements of set B.

- For the first element in A (let's say 1), there are 5 choices from set B.
- For the second element in A (let's say 2), there are 4 remaining choices from set B.
- For the third element in A (let's say 3), there are 3 remaining choices from set B.

Thus, the total number of one-one functions is calculated as:
\[ \text{Number of one-one functions} = 5 \times 4 \times 3 = 60 \]

### Step 3: Calculate the Number of Strictly Monotonic Functions
A strictly monotonic function can either be strictly increasing or strictly decreasing.

1. **Increasing Functions**:
- We need to select 3 distinct elements from set B to form an increasing function. The number of ways to choose 3 elements from 5 is given by the combination formula \( \binom{n}{r} \), which is \( \binom{5}{3} \).
- For each selection of 3 elements, there is exactly 1 way to arrange them in increasing order.

2. **Decreasing Functions**:
- Similarly, for decreasing functions, there is also exactly 1 way to arrange the selected 3 elements in decreasing order.

Thus, the total number of strictly monotonic functions (both increasing and decreasing) is:
\[ \text{Total monotonic functions} = 2 \times \binom{5}{3} \]
Calculating \( \binom{5}{3} \):
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
So,
\[ \text{Total monotonic functions} = 2 \times 10 = 20 \]

### Step 4: Calculate the Ratio
Now we can find the ratio of the number of one-one functions to the number of strictly monotonic functions:
\[ \text{Ratio} = \frac{\text{Number of one-one functions}}{\text{Total monotonic functions}} = \frac{60}{20} = 3 \]

### Final Answer
The ratio of the number of one-one functions to the number of strictly monotonic functions is **3**.

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