# Pascal's Rule

## Contents

## Theorem

Let $\dbinom n k$ be a binomial coefficient.

For positive integers $n, k$ with $1 \le k \le n$:

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

This is also valid for the real number definition:

- $\forall r \in \R, k \in \Z: \dbinom r {k - 1} + \dbinom r k = \dbinom {r + 1} k$

Thus the binomial coefficients can be defined using the following recurrence relation:

- $\dbinom n k = \begin{cases} 1 & : k = 0 \\ 0 & : k > n \\ \dbinom {n - 1} {k - 1} + \dbinom {n - 1} k & : \text{otherwise} \end{cases}$

### Complex Numbers

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer:

- $\dbinom z {w - 1} + \dbinom z w = \dbinom {z + 1} w$

where $\dbinom z w$ is a binomial coefficient.

## Direct Proof

Let $n, k \in \N$ with $1 \le k \le n$.

\(\displaystyle \binom n k + \binom n {k - 1}\) | \(=\) | \(\displaystyle \frac {n!} {k! \, \paren {n - k}!} + \frac {n!} {\paren {k - 1}! \, \paren {n - \paren {k - 1} }!}\) | $\quad$ Definition of Binomial Coefficient | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {n! \, \paren {n - \paren {k - 1} } } {k! \, \paren {n - k}! \, \paren {n - \paren {k - 1} } } + \frac {n! \, k} {\paren {k - 1}! \, \paren {n - \paren {k - 1} }! \ k}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {n! \, \paren {n - k + 1} } {k! \, \paren {n - k + 1}!} + \frac {n! \, k} {k! \, \paren {n - k + 1}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {n! \, \paren {n - k + 1} + n! \, k} {k! \, \paren {n - k + 1}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {n! \, \paren {n - k + 1 + k} } {k! \, \paren {n - k + 1}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {n! \, \paren {n + 1} } {k! \, \paren {n - k + 1}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\paren {n + 1}!} {k! \, \paren {n + 1 - k}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \binom {n + 1} k\) | $\quad$ Definition of Binomial Coefficient | $\quad$ |

$\blacksquare$

## Combinatorial Proof

Suppose you were a member of a club with $n + 1$ members (including you).

Suppose it were time to elect a committee of $k$ members from that club.

From Cardinality of Set of Subsets, there are $\dbinom {n + 1} k$ ways to select the members to form this committee.

Now, you yourself may or may not be elected a member of this committee.

Suppose that, after the election, you are not a member of this committee.

Then, from Cardinality of Set of Subsets, there are $\dbinom n k$ ways to select the members to form such a committee.

Now suppose you *are* a member of the committee. Apart from you, there are $k - 1$ such members.

Again, from Cardinality of Set of Subsets, there are $\dbinom n {k - 1}$ ways of selecting the *other* $k - 1$ members so as to form such a committee.

In total, then, there are $\dbinom n k + \dbinom n {k - 1}$ possible committees.

Hence the result.

$\blacksquare$

## Proof for Real Numbers

\(\displaystyle \left({r + 1}\right) \binom r {k - 1} + \left({r + 1}\right) \binom r k\) | \(=\) | \(\displaystyle \left({r + 1}\right) \binom r {k - 1} + \left({r + 1}\right) \binom r {r - k}\) | $\quad$ Symmetry Rule for Binomial Coefficients | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle k \binom {r + 1} k + \left({r - k + 1}\right) \binom {r + 1} {r - k + 1}\) | $\quad$ Factors of Binomial Coefficient | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle k \binom {r + 1} k + \left({r - k + 1}\right) \binom {r + 1} k\) | $\quad$ Symmetry Rule for Binomial Coefficients | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({r + 1}\right) \binom {r + 1} k\) | $\quad$ | $\quad$ |

Dividing by $\left({r + 1}\right)$ yields the result.

$\blacksquare$

## Also known as

Some sources give this as **Pascal's identity**.

## Also presented as

Some sources present this as:

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

## Also see

## Source of Name

This entry was named for Blaise Pascal.

## Sources

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of Mathematical Functions*... (previous) ... (next): $3.1.4$: Binomial Coefficients - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 19$: Theorem $19.10$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 3$: The Binomial Formula and Binomial Coefficients: $3.6$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.18$: Sequences Defined Inductively: Exercise $3 \ \text{(c)}$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.1$: The integers: Exercise $12$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $35$ - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients: $1.30$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $35$