First some background information:
T-cell receptors consist of two proteins called alpha and beta. All proteins are made of chains of amino acids. Each T-Cell receptor protein chain has a bottom part that is constant like the trunk of a tree and a top part that is divided into three major sections called Variable, Diversity, and Joining. Human DNA has the genetic code to make 109 different protein segments for the variable portion of the Alpha chain. Similarly there are 61 different possible Joining segments. The alpha chain has no Diversity segment. It has only one constant trunk segment, so to calculate the number of possible different alpha chains just multiply 109 x 61 x 1 = 6649
Here is a table to help you:
Try to calculate the total possible number of T-Cell receptors that your immune system can make. (Hint: multiply everything together.) How big is your number? Is it in the millions, billions, or trillions?
Now add this fact. A special enzyme puts a few extra amino acids between the Variable and Diversity chains. These are chosen randomly from our 21 amino acids, so you need to multiply again by 21 three or four times for each chain. Now how many possibilities are there?