The namesake theorem of these numbers describes their generation by expanding "one plus x" to the n, and they make up Pascal's Triangle.
Lucas's theorem reduces one of these numbers to a product of them modulo p. Products of these numbers can be summed using Vandermonde's convolution, and a sum of them can be simplified with the hockey-stick identity. If the row index is fixed, they sum to (*) "2 to the n," and they're given by the formula [read underlined part slowly] "n factorial over k factorial times (n minus k) factorial." The namesake theorem of these numbers describes their generation by expanding "one plus x" to the n, and they make up Pascal's Triangle. For 10 points, name these "coefficients" that describe the number of ways to choose k things from n possibilities, or n choose k.
Lucas's theorem reduces one of these numbers to a product of them modulo p. Products of these numbers can be summed using Vandermonde's convolution, and a sum of them can be simplified with the hockey-stick identity.
binomial coefficients (prompt on descriptive answers like “n choose k” or “members of Pascal’s triangle” or “number of combinations”)