## 1175. Prime Arrangements

Return the number of permutations of 1 to `n` so that prime numbers are at prime indices (1-indexed.)

(Recall that an integer is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers both smaller than it.)

Since the answer may be large, return the answer modulo `10^9 + 7`.

Example 1:

```Input: n = 5
Output: 12
Explanation: For example [1,2,5,4,3] is a valid permutation, but [5,2,3,4,1] is not because the prime number 5 is at index 1.
```

Example 2:

```Input: n = 100
Output: 682289015
```

Constraints:

• `1 <= n <= 100`

## Rust Solution

``````struct Solution;

impl Solution {
fn number_of_primes(n: usize) -> i32 {
let mut a: Vec<bool> = vec![true; n + 1];
a = false;
a = false;
let mut i: usize = 2;
while i * i <= n {
if a[i] {
let mut j: usize = 2;
while i * j <= n {
a[i * j] = false;
j += 1;
}
}
i += 1;
}
let mut res = 0;
for k in 0..=n {
if a[k] {
res += 1;
}
}
res
}

fn num_prime_arrangements(n: i32) -> i32 {
let primes = Self::number_of_primes(n as usize);
let mut product = 1i64;
for i in 1..=primes {
product *= i as i64;
product %= 1_000_000_007;
}
for i in 1..=(n - primes) {
product *= i as i64;
product %= 1_000_000_007;
}
product as i32
}
}

#[test]
fn test() {
assert_eq!(Solution::num_prime_arrangements(5), 12);
assert_eq!(Solution::num_prime_arrangements(100), 682_289_015);
}
``````

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