To solve the problem of finding the number of ordered 5-tuples \((u, v, w, x, y)\) where \(u, v, w, x, y \in [1, 11]\) that satisfy the inequality
\[
2^{\sin^2 u + 3 \cos^2 v} \cdot 3^{\sin^2 w + \cos^2 x} \cdot 5^{\cos^2 y} \geq 720,
\]
we can follow these steps:
### Step 1: Factor the number 720
First, we can express 720 in terms of its prime factors:
\[
720 = 2^4 \cdot 3^2 \cdot 5^1.
\]
### Step 2: Set up the inequality
We can rewrite the inequality as:
\[
2^{\sin^2 u + 3 \cos^2 v} \cdot 3^{\sin^2 w + \cos^2 x} \cdot 5^{\cos^2 y} \geq 2^4 \cdot 3^2 \cdot 5^1.
\]
This implies:
\[
\sin^2 u + 3 \cos^2 v \geq 4,
\]
\[
\sin^2 w + \cos^2 x \geq 2,
\]
\[
\cos^2 y \geq 1.
\]
### Step 3: Analyze each component
1. **For \( \cos^2 y \geq 1 \)**:
- This means \( \cos^2 y = 1 \), which occurs when \( y = 1, 11 \) (since \( \cos^2 y = 1 \) at these points). Thus, there are **2 choices** for \(y\).
2. **For \( \sin^2 u + 3 \cos^2 v \geq 4 \)**:
- The maximum value of \(\sin^2 u\) is 1 (when \(u = 1, 11\)), and the maximum value of \(3 \cos^2 v\) is 3 (when \(v = 1, 11\)). Thus, we need:
\[
\sin^2 u + 3 \cos^2 v \geq 4 \implies \sin^2 u = 1 \text{ and } \cos^2 v = 1.
\]
- This occurs when \(u = 1, 11\) and \(v = 1, 11\). Thus, there are **2 choices** for \(u\) and **2 choices** for \(v\).
3. **For \( \sin^2 w + \cos^2 x \geq 2 \)**:
- The maximum value of \(\sin^2 w\) is 1 and the maximum value of \(\cos^2 x\) is also 1. Thus, we need:
\[
\sin^2 w + \cos^2 x = 2 \implies \sin^2 w = 1 \text{ and } \cos^2 x = 1.
\]
- This occurs when \(w = 1, 11\) and \(x = 1, 11\). Thus, there are **2 choices** for \(w\) and **2 choices** for \(x\).
### Step 4: Calculate the total combinations
Now, we can calculate the total number of ordered 5-tuples:
\[
\text{Total} = (\text{choices for } u) \cdot (\text{choices for } v) \cdot (\text{choices for } w) \cdot (\text{choices for } x) \cdot (\text{choices for } y) = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5 = 32.
\]
### Step 5: Conclusion
Thus, the total number of ordered 5-tuples \((u, v, w, x, y)\) that satisfy the inequality is:
\[
\boxed{32}.
\]
