MHB Optimizing Binomial Coefficients for Maximum Value

AI Thread Summary
The discussion focuses on determining the value of k that maximizes the term A_k in the binomial expansion of (1 + 1/5)^1000, where A_k = (1000 choose k)(1/5)^k. Through analysis, it is concluded that k should satisfy the inequality derived from comparing consecutive terms, leading to the condition 1/(1000-k) > 1/(5k+5). This simplifies to find that k = 166 is the optimal solution for maximizing A_k. The participants express appreciation for the collaborative effort in reaching this conclusion. The thread emphasizes the mathematical reasoning behind optimizing binomial coefficients.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
From the binomial theorem, we have

$\displaystyle \begin{align*}\left(1+\dfrac{1}{5}\right)^{1000}&={1000 \choose 0}\left(\dfrac{1}{5}\right)^{0}+{1000 \choose 1}\left(\dfrac{1}{5}\right)^{1}+{1000 \choose 2}\left(\dfrac{1}{5}\right)^{2}+\cdots+{1000 \choose 1000}\left(\dfrac{1}{5}\right)^{1000}\\&=A_0+A_1+A_2+\cdots+A_{1000} \end{align*}$

where $\displaystyle A_k={1000 \choose k}\left(\dfrac{1}{5}\right)^{k}$.

For which $k$ is $A_k$ the largest?
 
Mathematics news on Phys.org
anemone said:
From the binomial theorem, we have

$\displaystyle \begin{align*}\left(1+\dfrac{1}{5}\right)^{1000}&={1000 \choose 0}\left(\dfrac{1}{5}\right)^{0}+{1000 \choose 1}\left(\dfrac{1}{5}\right)^{1}+{1000 \choose 2}\left(\dfrac{1}{5}\right)^{2}+\cdots+{1000 \choose 1000}\left(\dfrac{1}{5}\right)^{1000}\\&=A_0+A_1+A_2+\cdots+A_{1000} \end{align*}$

where $\displaystyle A_k={1000 \choose k}\left(\dfrac{1}{5}\right)^{k}$.

For which $k$ is $A_k$ the largest?

I looked for k such that $\frac{1000!}{k!(1000-k)!5^k}>\frac{1000!}{(k+1)!(999-k)!5^{k+1}}$.

This simplifies to $\frac{1}{1000-k}>\frac{1}{5k+5}$.

Leading to k=166 if I didn't make any dopey errors.
 
Thanks M R for your participation and your correct solution!(Yes)

A solution proposed by other:

Note that $\displaystyle A_k={1000 \choose k}\left(\dfrac{1}{5}\right)^k=\dfrac{1000!}{5^k(k!)(1000-k)!}$, so to maximize $A_K$ means we must minimize its denominator, and if we let it as $P_k=5^k(k!)(1000-k)!$, for all $k$, we have $P_{k+1}=5^{k+1}((k+1)!)(1000-(k+1))!=5^{k+1}((k+1)!)(999-k)!=P_k\left(\dfrac{5(k+1)}{1000-k}\right)$.

For small values of $k$, $\dfrac{5(k+1)}{1000-k}<1$, so we must have $P-0>P_1>P_2>\cdots$. The minimum value will thus come at the smallest value of $k$ for which $\dfrac{5(k+1)}{1000-k}>1$, so we must have $5(k+1)>1000-k\,\,\rightarrow k=166$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top