Monsieur Pierre de Fermat,
I hope that you are in good health and are continuing to make leaps and bounds in your impressive mathematical work. I cannot believe the year 1653 is almost over, it has been a busy few year since the passing of my father. But do not worry, I have not yet given up my work in science and mathematics! In my studies of binomial coefficients I have organized the data in a way that shows some interesting patterns! You can create a triangle by creating rows as follows.
You now continue down the triangle, with each inside entry being the sum of the two nearest entries above it. If you continue this method for 6 rows you get.
Beyond finding the coefficients of binomials, I find this triangle to have uses in the theories of counting and probability. If you are looking for the number of possibilities to draw objects from a group you can go down to the row of the number you are choosing from (The top row counts as 0) and then go to the right the number you are choosing.
Kindest regards,
Blaise Pascal
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