FMM17

Friday Morning Math Problem

A Binomial Problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \binom{16}{r} = \binom{16}{2r+1} }

Find r.

Challenge problem:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \binom{120}{r} = \binom{120}{3r+4} }

Find r.

Solution

First, if we solve it naively, we get r = 2r + 1 -r = 1 r = -1 We use the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \binom{n}{m} = \binom{n}{n-m}} 2r + 1 =16 - (2r+1) r = 16 - (2r+1) r = 16 - 2r - 1 3r = 15 r = 5 What more do we need to do? We can use this identity to transform r = 2r + 1, but there may also exist other integers 0 <= r < s <= n such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \binom{n}{r} = \binom{n}{s}} We can prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \binom{n}{r} = \binom{n}{s}} only holds if r = s or r = n - s This is something we may implicitly assume, due to our own understanding of the nature of the binomial numbers and Pascal's Triangle. But to be rigorous we have to cover all our bases. Challenge problem: Solving naively we get r = 3r + 4 -4r = 4 r = -1 To transform, use the same identity as above 3r + 4 = r 3r + 4 = 120 - r 4r = 120 - 4 r = (120 - 4)/4 = 29