Binomial Coefficient, Pascal's Triangle and Newton's Binomial

SAT Questions that focus on Binomial Coefficient, Pascal's Triangle and Newton's Binomial require knowledge of the following topics.

Factorial

The factorial of a positive integer "n", denoted by n!, is defined as: $n!=n(n-1)(n-2).....1$

For example,

$2!=2*1=2$

$3!=3*2*1=6$

$4!=4*3*2*1=24$

Binomial Coefficient

The binomial coefficient is written as $\binom{n}{p}$, where "n" and "p" are non negative integers, and "n" is greater than "p".

By definition, the binomial coefficient is given by the formula:

$\binom{n}{p}=\frac{n!}{p!(n-p)!}$

For example, $\binom{4}{2}=\frac{4!}{2!(4-2)!}=\frac{24}{2.2}=6$

Pascal's Triangle

Pascal's triangle is a triangular array that displays all binomial coefficients:

Row 0: $\binom{0}{0}$

Row 1: $\binom{1}{0}$ $\binom{1}{1}$

Row 2: $\binom{2}{0}$ $\binom{2}{1}$ $\binom{2}{2}$

Row 3: $\binom{3}{0}$ $\binom{3}{1}$ $\binom{3}{2}$ $\binom{3}{3}$

......................


If we develop these binomial coefficients, we have the numerical form of the Pascal' Triangle:

Row 0: $1$
Row 1: $1$ $1$
Row 2: $1$ $2$ $1$
Row 3: $1$ $3$ $3$ $1$
Row 4: $1$ $4$ $6$ $4$ $1$
..............................

Consider now the following binomial expansions:

$(x+y)^2=1x^2+2xy+1y^2$
$(x+y)^2=1x^2y^0+2x^1y^1+1x^0y^2$

and

$(x+y)^3=1x^3-3x^2y+3xy^2-1y^3$
$(x+y)^3=1x^3y^0-3x^2y^1+3x^1y^2-1x^0y^3$

Notice that the coefficients in the expansion of $(x+y)^2$ are the numbers in row two of Pascal's triangle (1, 2 and 1), and the coefficients in the expansion of $(x+y)^3$ are the numbers in row three of Pascal's triangle (1, 3, 3 and 1).

In general, the expansion of the binomial $(x+y)^n$ (where "n" is a positive integer) is given by the formula:

$(x+y)^n=a_0x^ny^0+a_1x^{n−1}y^1+a_2x^{n−2}y^2+...+a_{n−1}x^1y^{n−1}+a_nx^0y^n$,
where the coefficients $a_k$ are the numbers on row "n" of Pascal's triangle. In other words,

$a_k=\binom{n}{k}$

Therefore, the binomial expansion can be writen as:

$(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k$

This is the Binomial Theorem (or Binomial Expansion, or Newton's Binomial).

Sum of the elements in a Pascal's Triangle row

The sum of the elements in row "n" of the Pascal's Triangle is given by the formula: $2^n$.

Row 0: $1=2^0$
Row 1: $1+1=2=2^1$
Row 2: $1+2+1=4=2^2$
Row 3: $1+3+3+1=8=2^3$
Row 4: $1+4+6+4+1=16=2^4$

Solved SAT Practice Tests

Find Practice Tests in the following link: Not Available