# 377. Combination Sum IV

## The Problem

Link to original problem on Leetcode.

Given an array of **distinct** integers `nums`

and a target integer `target`

, return the *number of possible combinations that add up to* `target`

.

The test cases are generated so that the answer can fit in a **32-bit** integer.

## Examples

Example 1:

```
Input: nums = [1,2,3], target = 4
Output: 7
Explanation:
The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)
Note that different sequences are counted as different combinations.
```

Example 2:

```
Input: nums = [9], target = 3
Output: 0
```

## Constraints

`1 <= nums.length <= 200`

`1 <= nums[i] <= 1000`

- All the elements of
`nums`

are**unique**. `1 <= target <= 1000`

## My Solution

### Dynamic Programming - Bottom-up

Here we've got another problem solved with dynamic programming. We can break apart this problem into smaller sub-problems to get our solution.

In this first variation, we use the bottom-up approach. First, create our cache `dp`

as an array with length of `target + 1`

. We'll initialize `dp[0] = 1`

, as there's exactly one combination to reach a target of zero (i.e., use no numbers). Next we loop over the range `0`

to `target - 1`

. If we have no value for `dp[i]`

, we continue to the next number in the range. Since we initialized `dp[0]`

to `1`

, we'll at least run the rest of the loop that first time if `target > 0`

.

Then for each number `i`

in the range, we'll loop over each `num`

in the `nums`

array. If `num + i <= target`

, then we've found a possibly valid path toward the target. So we'll set `dp[num + i]`

equal to itself plus `dp[i]`

. That is to say, we take the number of combinations we know gets us to `dp[i]`

and add them to any other already known number of combinations that get us to `num + i`

.

Once we've finished all our loops, the total number of valid combinations will be at index `target`

in `dp`

, so we return `dp[target]`

. This is true even if the `target`

is `0`

, since we initialized `dp[0] = 1`

.

`function combinationSum4(nums: number[], target: number): number {`

let dp = Array.from({length: target + 1}, () => 0);

dp[0] = 1;

for (let i = 0; i < target; i++) {

if (!dp[i]) continue;

for (let num of nums) {

if (num + i <= target) {

dp[i + num] += dp[i];

}

}

}

return dp[target];

};

The time complexity is $O(n * m)$, where $n$ is the `target`

and $m$ is the length of `nums`

. Space complexity is $O(n)$.

### Dynamic Programming - Top-down

The top-down approach is nearly identical to the bottom-up approach. Here we loop not from `0`

through `target - 1`

, but instead from `1`

through `target`

. We don't need to check if `dp[i]`

exists here before running the inner loop, because we'll be setting values to `dp[i]`

instead of `dp[num + i]`

. Next we loop again over `num`

of `nums`

. If `num <= i`

(i.e., there is some value less than `i`

and greater than or equal to `0`

that, added to `num`

, gets us to target of `i`

), we'll set `dp[i]`

to itself plus the combinations we've observed at `dp[i - num]`

. And again, we return `dp[target]`

.

Just like with bottom-up, time complexity is $O(n * m)$, where $n$ is the `target`

and $m$ is the length of `nums`

. Space complexity is $O(n)$.

`function combinationSum4(nums: number[], target: number): number {`

let dp = Array.from({length: target + 1}, () => 0);

dp[0] = 1;

for (let i = 1; i <= target; i++) {

for (let num of nums) {

if (num <= i) {

dp[i] += dp[i - num];

}

}

}

return dp[target];

};