Question

What is the row of Pascal’s triangle containing the binomial coefficients (nk),0≤k≤9?

Binomial probability
ANSWERED
asked 2021-01-22
What is the row of Pascal’s triangle containing the binomial coefficients \(\displaystyle{\left({n}{k}\right)},{0}≤{k}≤{9}?\)

Answers (1)

2021-01-23

I've been considering entry i in row n of Pascal's Triangle's Triangle, so for \(\displaystyle{U}\le{i}\le{n}\), we have
\(\displaystyle{\left({n}{i}\right)}={n}\frac{!}{{{i}!{\left({n}-{i}\right)}!}}\)
\(\rightarrow\) The row of (n k) are the binomial coefficients (n k) evaluated at
\(k=0,1,2,3,4,5,6,7,8,9\)
\(\displaystyle{\left({n}{0}\right)}={n}\frac{!}{{{0}!{\left({n}-{0}\right)}!}}\)
\(\displaystyle{\left({n}{1}\right)}={n}\frac{!}{{{1}!{\left({n}-{1}\right)}!}}\)
\(\displaystyle{\left({n}{2}\right)}={n}\frac{!}{{{2}!{\left({n}-{2}\right)}!}}\)
\(\displaystyle{\left({n}{3}\right)}={n}\frac{!}{{{3}!{\left({n}-{3}\right)}!}}\)
\(\displaystyle{\left({n}{4}\right)}={n}\frac{!}{{{4}!{\left({n}-{4}\right)}!}}\)
\(\displaystyle{\left({n}{5}\right)}={n}\frac{!}{{{5}!{\left({n}-{5}\right)}!}}\)
\(\displaystyle{\left({n}{6}\right)}={n}\frac{!}{{{6}!{\left({n}-{6}\right)}!}}\)
\(\displaystyle{\left({n}{7}\right)}={n}\frac{!}{{{7}!{\left({n}-{7}\right)}!}}\)
\(\displaystyle{\left({n}{8}\right)}={n}\frac{!}{{{8}!{\left({n}-{8}\right)}!}}\)
\(\displaystyle{\left({n}{9}\right)}={n}\frac{!}{{{9}!{\left({n}-{9}\right)}!}}\)
I've been considering entry i in row n of Pascal's Triangle, so for \(\displaystyle{0}\le{i}\le{n}\), we have
\(\displaystyle{\left({n}{i}\right)}={n}\frac{!}{{{i}!{\left({n}-{i}\right)}!}}\)

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