Answer:
n = 45 and r = 19
Step-by-step explanation:
To find integers n and r such that
[tex]\displaystyle \binom{43}{17}+2\binom{43}{18} + \binom{43}{19}=\binom{n}{r}[/tex]
we can use Pascal's Identity, which is a combinatorial identity that relates binomial coefficients:
[tex]\displaystyle \binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}[/tex]
Begin by rewriting the second term as the sum of two identical terms:
[tex]\displaystyle \binom{43}{17} + \binom{43}{18} + \binom{43}{18} + \binom{43}{19}=\binom{n}{r}[/tex]
Now apply Pascal's Identity repeatedly to simplify the given expression:
[tex]\displaystyle \binom{44}{18} + \binom{43}{18} + \binom{43}{19}=\binom{n}{r}[/tex]
[tex]\displaystyle \binom{44}{18} + \binom{44}{19}=\binom{n}{r}[/tex]
[tex]\displaystyle \binom{45}{19}=\binom{n}{r}[/tex]
Therefore, n = 45 and r = 19.
