Pascal's triangle is a visual representation of the coefficients involved in a binomial expansion. The [tex]n[/tex]th row of the triangle gives the coefficients of the terms in the expansion of [tex](a+b)^{n-1}[/tex].
The triangle itself looks like
[tex]\begin{matrix}1\\1&1\\1&2&1\\1&3&3&1\end{matrix}[/tex]
and so on, while the expansions for [tex]n=1,2,3,4[/tex] are
[tex](a+b)^{1-1}=(a+b)^0=1[/tex]
[tex](a+b)^{2-1}=(a+b)^1=1a+1b[/tex]
[tex](a+b)^{3-1}=(a+b)^2=1a^2+2ab+1b^2[/tex]
[tex](a+b)^{4-1}=(a+b)^3=1a^3+3a^2b+3ab^2+1b^3[/tex]
and so on.
The binomial theorem says that
[tex](a+b)^{n-1}=\displaystyle\sum_{k=0}^{n-1}\binom{n-1}ka^{n-1-k}b^k[/tex]
[tex](a+b)^{n-1}=\dbinom{n-1}0a^{n-1}b^0+\dbinom{n-1}1a^{n-2}b^1+\cdots+\dbinom{n-1}{n-2}a^1b^{n-2}+\dbinom{n-1}{n-1}a^0b^{n-1}[/tex]
[tex](a+b)^{n-1}=\dbinom{n-1}0a^{n-1}+\dbinom{n-1}1a^{n-2}b+\cdots+\dbinom{n-1}{n-2}ab^{n-2}+\dbinom{n-1}{n-1}b^{n-1}[/tex]
The numbers in the [tex]n[/tex]th row of the triangle are just [tex]\dbinom{n-1}k[/tex], with [tex]k=0,1,\ldots,n-2,n-1[/tex].
So no, the binomial theorem and Pascal's triangle are not the same thing. Pascal's triangle is a way of organizing the pattern exhibited by the result of the binomial theorem.
