Q 7: What is the number of `Hello's printed by the pseudo code below? (for i from lo to hi

exhaust i between lo and hi inclusive, and is a empty loop when lo is greater than hi)

(a) for i from 1 to n
for j from 1 to i - 1
for k from 1 to j - 1
print `Hello'

(b) for i from 1 to 10
for j from i to 10
for k from i to j
print `Hello'

Assuming that k from 1 to n fails to execute if n < 1,

when i=1, j fails
when i=2, k fails at j=1
when i=3, j=1..2, k=0+1..1=1
when i=4, j=1..3, k=0+1+2=3
when i=5, j=1..4, k=0+1+2+3=6
...
when i=n, j=1..n-1, k=0+1+2+...+n-2 = (n-2)(n-1)/2

so, the total prints executed is
1+3+...+(n-2)(n-1)/2
= (n-2)(n-1)(n)/6

now do a similar analysis on (b)

A: For the pseudo code (a), the number of `Hello's printed can be calculated as follows:

- For each value of `i` from 1 to n, there will be `i - 1` values of `j` between 1 and `i - 1`.
- For each value of `j`, there will be `j - 1` values of `k` between 1 and `j - 1`.
- Therefore, the total number of `Hello's printed can be calculated as the sum of `i - 1` for each value of `i` from 1 to n.

The equation for the sum of integers from 1 to n is given by (n * (n + 1)) / 2. However, in this case, we need to subtract 1 from each value of `i` before calculating the sum.

So, the total number of `Hello's` printed in pseudo code (a) is:
((n * (n + 1)) / 2) - n.

For the pseudo code (b), the number of `Hello's` printed can be calculated as follows:

- For each value of `i` from 1 to 10, there will be 10 - `i` + 1 values of `j` between `i` and 10.
- For each value of `j`, there will be `j` - `i` + 1 values of `k` between `i` and `j`.
- Therefore, the total number of `Hello's` printed can be calculated as the sum of (`j` - `i` + 1) for each value of `j` from `i` to 10 and each value of `i` from 1 to 10.

The nested loops in pseudo code (b) create a triangular pattern where the number of `Hello's` printed increases with each iteration.

I hope that helps!

(a) To determine the number of `Hello's printed by the pseudo code, we need to count the number of times the innermost loop executes (the `print `Hello'` statement).

In this case, the innermost loop is:

```
for k from 1 to j - 1
print `Hello'
```

The value of `j` is controlled by the middle loop:

```
for j from 1 to i - 1
for k from 1 to j - 1
print `Hello'
```

And the value of `i` is controlled by the outer loop:

```
for i from 1 to n
for j from 1 to i - 1
for k from 1 to j - 1
print `Hello'
```

Therefore, the number of `Hello's` printed by the pseudo code is dependent on the value of `n`. If `n` is known, we can substitute the value of `n` into the outer loop and calculate the number of `Hello's`.

(b) Similarly, for the second pseudo code snippet:

```
for i from 1 to 10
for j from i to 10
for k from i to j
print `Hello'
```

The number of `Hello's` printed is also dependent on the value of `i`. If `i` is known, we can substitute the value of `i` into the outer loop and calculate the number of `Hello's`.

To determine the number of "Hello's" printed by the given pseudo code, you need to count the number of times the `print `Hello'` statement is executed.

For (a), the nested loops iterate according to the following ranges:
- The outer loop `i` iterates from 1 to `n`.
- The middle loop `j` iterates from 1 to `i - 1`.
- The innermost loop `k` iterates from 1 to `j - 1`.

Each time the innermost loop executes, it prints "Hello". Therefore, to calculate the total number of "Hello's" printed, you need to find the total number of iterations of the innermost loop.

Let's break it down:
- For `j` = 1, the innermost loop doesn't execute.
- For `j` = 2, the innermost loop executes once.
- For `j` = 3, the innermost loop executes twice.
- For `j` = 4, the innermost loop executes three times.
- And so on...

The total number of "Hello's" printed can be calculated using the summation formula:

```
Total = 1 + 2 + 3 + ... + (n - 2) + (n - 1)
```

This is a sum of the first `(n - 1)` natural numbers, which can be simplified using the formula for the sum of an arithmetic series:

```
Total = ((n - 1) * n) / 2
```

For (b), the nested loops iterate according to the following ranges:
- The outer loop `i` iterates from 1 to 10.
- The middle loop `j` iterates from `i` to 10.
- The innermost loop `k` iterates from `i` to `j`.

The innermost loop executes `(j - i + 1)` times. Therefore, to calculate the total number of "Hello's" printed, you need to find the total number of iterations of the innermost loop for all combinations of `i` and `j` within their respective ranges.

Let's break it down:
- When `i` = 1, the innermost loop executes 10 times.
- When `i` = 2, the innermost loop executes 9 times.
- When `i` = 3, the innermost loop executes 8 times.
- And so on...

The total number of "Hello's" printed can be calculated using the summation formula:

```
Total = 10 + 9 + 8 + ... + 2 + 1
```

This is a sum of the first 10 natural numbers, which can be again simplified using the formula for the sum of an arithmetic series:

```
Total = (10 * 11) / 2 = 55
```

Therefore, the number of "Hello's" printed in (a) is given by the formula `((n - 1) * n) / 2`, and in (b) is 55.