Paramedic Permutations

Warning: Math Ahead

Paramedic Permutations
Rabbits are apparently quite good at multiplication

Math was never my most favorite subject in school. I struggled with the concept of negative numbers, multiplications of negatives, and in particular expanding equations/polynomials. I felt dumb, until in college I was able to take a statistics course that was more centered in the business world, particularly in how to get practical results using modern tools like Excel.

That course had a great impact on my confidence, and I re-approached math through the guise of statistics and programming. I may still struggle with finding slope intercepts and putting it into the right notation, but I am at least more in tune with mean, medians, mode, ranges, standard deviations, etc.

If you were told there would be no math on my posts, even if I hate it, you would have been told incorrectly.


Mike Herbert of RQI Partners called me the other day to tell me he had heard on a podcast (the title of which I unfortunately failed to write down) about first responders and permutations. It's a twist of why I once reached out to his company in the first place.

In early 2021, I was preparing a presentation submission for a conference on EMS CPR training post COVID, and I was hitting against a wall. Looking at our numbers, we were showing about a 20-30% turnover in staff each year. It varied from position to position, but after reaching out to some experts we found that what we were seeing mirrored nationwide trends.[1]

Scott Moore of the American Ambulance Association in his 2022 report on turnover put it very succinctly.

EMS agencies are experiencing a full turnover of all staff every 3-4 years. Greater than one third of all new hires, turnover within their first year of employment.[2]

Every two years, crews would sit together in a room and re-certify their CPR and/or ACLS/PALS courses. That was the extent of required non-real-world trainings many providers would get. Sure, shifts would hold some training to integrate new hires, but as far as formal, measured, and validated training, 2 was about it, and there was a chance entire new hires went without training formally as part of a larger team.

Permutations, Factorials, and Combinations

If A = Every year I have 30% turnover, B = Every 3 Years I potentially have 100% new provider pairings, and C = I am running a formal CPR related training every 2 years, solve for why I am running a system that isn't geared towards improvement.

Enter Permutations, Factorials, and Combinations.

You may be familiar with Factorials, which show you combinations in the format of 10!, where it multiples 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 to get an answer of 3,628,800. This is the possibility of all combinations of 10 given items that are positive integers.

What Mike helped me realize with his call was that there is also something called permutations, which vary slightly from factorials in that order matters. While the answer would be the same, a permutation is a way of arranging objects in a specific order. For example, if you have a team of 3 paramedics and EMTs, the order in which they are assigned to the team matters. So, the permutation of the team "John, Mary, Peter" is different from the permutation "Mary, John, Peter".

We can use factorials to calculate the number of possible permutations of the team in a more manageable way.

The formula to find this is nPr = n! / (n - r). In this, nPr is the number of permutations of a sample, n = 50 (total number of paramedics and EMTs), and r = 3 (number of people to be selected).

The number of possible permutations of a team of 3 people from a pool of 50 people is set up using this equation:

nPr = 50! / (50 - 3)!

or, rewritten

nPr = 50! / 47!

Now when that's expanded, it becomes

nPr = (50 x 49 x 48 x 47!) / 47!

Why the 47!? Well it represents the 47 x 46 x 45 all the way down to 1, but since there is a 47! in the denominator of (remember, 3 person team) they cancel out and you just need the 50 x 49 x 48.

This means that there are 117,600 possible permutations of the team of 3 paramedics and EMTs. It can be narrowed down with more advanced math to correctly assume when a Paramedic is team leader, more accurately tracking crews, but that number gave me a good outline of the problem I had with 117,600 possible combinations (with 30% of those turning over every year)

Validation Training

So no surprise, we wound up adopting RQI and RQI Teams to help solve this problem. I'll let Mike and his team make the true sales pitch, but the idea that, quarterly, I can send different combinations of teams to train together on a simulated cardiac arrest (using validation from the smart systems inside the mannequin and a video debrief of their performance) was a load off my mind.

It made me curious to where permutations could pop up other places an influence CPR effectiveness too.

In bystander-related trainings, it's a little easier when you have one person doing CPR and AED skills. Thats a permutation of, well, 1. But in real-life scenarios, if you have response teams or teams of bystanders, training that teamwork may need to be a more managed skill.

Take corrective action, for example. I would be willing to bet that many of us early in our careers would be unwilling to tell someone we didn't have a strong familiarity with to push harder, faster, or deeper during CPR. Yet, we take for granted that this will happen in CPR trainings daily. It wasn't until I had an automated mannequin tell me my mistakes that I caught my own individual performance- who knows what team-based performance we had missed in the interim.

Validated training is one tool that will address the permutational problems caused by regular EMS turnover. It's one I'm glad we've adopted, as outside of simple cardiac arrest improvement it also is a worthwhile tool to build familiarity and teamwork among ever-changing staff. It has it's usefulness in other world applications too, and I look forward to seeing others address the problem as well.

This will be the last Monday Feature for about a month- I need to take a lot of what I've written and synthesize it into a good 45-60 minute presentation, and I'm hoping to use the time I have been writing to do just that. If I finish sooner (or have something important to say ASAP) I'll surprise you with an early return, but until then look forward to regular weekly "Four Things for Friday" posts.

Take care and stay safe!

  1. ↩︎

  2. Ibid. ↩︎