Question 1
By 
[
𝑛
]
[n]we denote the set 
{
1
,
…
,
𝑛
}
{1,…,n}. A function 
𝑓
:
[
𝑚
]
→
[
𝑛
]
f:[m]→[n] is called monotone if 
𝑓
(
𝑖
)
≤
𝑓
(
𝑗
)
f(i)≤f(j)whenever 
𝑖
<
𝑗
i<j. Let 
𝑆
(
𝑚
,
𝑛
)
S(m,n) be the number of monotone injective functions from 
[
𝑚
]
[m] to 
[
𝑛
]
[n]. What is the number of monotone injective functions from 
[
10
]
[10] to 
[
20
]
[20]?

We are asked to find the number of monotone injective functions from \([10]\) to \([20]\), where \([n]\) is the set \(\{1, 2, \dots, n\}\), and a function \(f: [m] \to [n]\) is monotone if for any \(i < j\), \(f(i) \leq f(j)\). Additionally, the function is injective, meaning \(f(i) \neq f(j)\) for all \(i \neq j\).

### Understanding the problem

1. The function \(f\) is **monotone** and **injective**, meaning it must map smaller inputs to smaller outputs in a strictly increasing manner. 
2. This implies that the function is a strictly increasing function, i.e., \(f(i) < f(j)\) whenever \(i < j\).
3. The number of monotone injective functions from \([m]\) to \([n]\) is equivalent to choosing \(m\) distinct elements from \([n]\) and then ordering them in increasing order (which is done automatically by the monotonicity requirement).

### Step-by-step solution

The problem now reduces to counting how many ways we can choose 10 elements from the set \([20]\) (since the monotonicity enforces a strict order).

This is a simple **combinatorics problem** of selecting 10 elements from a set of 20, which is given by the binomial coefficient:

\[
S(10, 20) = \binom{20}{10}
\]

### Calculate the binomial coefficient

The formula for the binomial coefficient is:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

In this case:

\[
\binom{20}{10} = \frac{20!}{10! \cdot 10!}
\]

We can calculate this value.

\[
\binom{20}{10} = 184,756
\]

### Conclusion

The number of monotone injective functions from \([10]\) to \([20]\) is:

\[
S(10, 20) = 184,756
\]

Let me know if you'd like further details or have any questions!

Here are 5 related questions for further exploration:
1. How do you calculate the number of injective functions if monotonicity is not required?
2. What is the number of monotone injective functions from \([5]\) to \([15]\)?
3. How does the binomial coefficient change if you increase the size of the sets?
4. How can you generalize this concept for non-injective monotone functions?
5. What is the significance of monotone functions in real-world applications?

**Tip:** Binomial coefficients often arise in combinatorics, especially when counting ways to choose items from a set.

By [n] we denote the set {1,…,n}. A function f: [m]→[n] is called monotone if f(i)≤f(j) whenever i<j. Let S(m,n) be the number of monotone injective functions from [m] to [n]. What is the number of monotone injective functions from [10] to [20]?

Explore how to count the number of monotone injective functions from [10] to [20] using combinatorics. Learn how the binomial coefficient formula is applied to find the solution, with a detailed explanation. This problem is ideal for undergraduate math students.

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