The Science
Chosen structure:
All 3 exercises are easy to visualize, commonly practiced, and recognised as being some of the most essential for good health, muscle tone, and physique.
Push-ups have been chosen first to get the hardest exercise out of the way.
Squats are done second to provide a buffer between the two most demanding exercises.
Planks are done last as core training is gruelling. But once it's done, it's done.
Chosen exercises:
The chosen exercises are all calisthenics movements. Calisthenics is effective as it is one of the best ways to increase muscle strength and provides the best muscle definition for each unique body type.
Common names for major muscle groups:
Push-ups:
Pushups train the whole upper body and arms, mainly focusing on the chest and triceps, but can also train your shoulders, and core.
Push-ups give a great toning and square up your chest, and are a highly effective method of training the tricep, which makes up the seventy-five percent of the arm.
Regular push-ups train the pectoralis major and minor of the chest, and Elevated push-ups trainly the pectorals even greater, whereas diamond push-ups focus on training the triceps.
Other muscle groups trained are the shoulders (deltoids), biceps, abdominals, middle and lower back, and even leg muscles such as glutes and hamstrings.
The push-up provides conditioning for a large variety of muscles and stabilizers, which is why it's such a quintessential movement.
Squats:
The Squat exercise mainly targets the thighs (quadriceps & hamstrings), and the glutes (buttocks muscles).
Squats also train core-strength & stability, ankle mobility, back muscles, and calves. Squats are a quintessential movement for a healthy lower body.
Planks:
Your core is very important; a strong trunk makes for a strong tree, which allows for strong branches, etc.
Planks offer training to a wide range of muscle groups, and can give a full body workout in just one exercise.
Planks focus primarily on the core muscles in your abdominals, and the lower back (rectus abdominis, transverse abdominis and erector spinae).
Planks also exert work on your shoulders, chest, triceps, biceps, and to a lesser extent, your back and legs too.
By supporting your weight in planks it will allow you to achieve a more balanced body, which is important, as an imbalance in your muscles leaves a body prone to injury.
Conclusion
All three of these exercises aim to train the entire body, and should be implemented into any training routine for a healthier and stronger body and mind.