TAOCP 1.2.6 Exercise 5
The rows of Pascal's triangle give the coefficients of the expansion of $(x+y)^n$ according to the binomial theorem, Eq.
Section 1.2.6: Binomial Coefficients
Exercise 5. [05] Use Pascal's triangle to explain the fact that $11^4 = 14641$.
Verified: yes
Solve time: 41s
The rows of Pascal's triangle give the coefficients of the expansion of $(x+y)^n$ according to the binomial theorem, Eq. (13). The fourth row (corresponding to $n=4$) is $1, 4, 6, 4, 1$. Substituting $x = 10$ and $y = 1$, we have
= \binom{4}{0}10^0 + \binom{4}{1}10^1 + \binom{4}{2}10^2 + \binom{4}{3}10^3 + \binom{4}{4}10^4.$$ Evaluating each term, we find $$\binom{4}{0}10^0 = 1,\qquad \binom{4}{1}10^1 = 4 \cdot 10 = 40,\qquad \binom{4}{2}10^2 = 6 \cdot 100 = 600,$$$$\binom{4}{3}10^3 = 4 \cdot 1000 = 4000,\qquad \binom{4}{4}10^4 = 1 \cdot 10000 = 10000.$$ Summing these contributions gives $$1 + 40 + 600 + 4000 + 10000 = 14641.$$Hence $11^4 = 14641$ follows directly from the coefficients in Pascal's triangle. $\boxed{14641}$ ∎