Pascal's Rule/Also presented as

Pascal's Rule: Also presented as

Some sources present Pascal's rule as:

$\dbinom n k + \dbinom n {k + 1} = \dbinom {n + 1} {k + 1}$

Others present it in the form:

$\dbinom {n - 1} {k - 1} + \dbinom {n - 1} k = \dbinom n k$

Sources

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Two important relations
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis: Theorem $19.10$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.6$: Theorem $8: \ 3$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 3$: The Binomial Formula and Binomial Coefficients: Properties of Binomial Coefficients: $3.6$
1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-1}$ Permutations and Combinations: Exercise $3$
1972: George Pólya and Gábor Szegő: Problems and Theorems in Analysis I (Eng. ed.): $\S 1.2$: Problem $31.1$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: Exercise $12$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $35$
1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients: $1.30$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text{D} \ (9)$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 3$: The Binomial Formula and Binomial Coefficients: Binomial Coefficients: $3.6.$