Zadanie 4.63.
Wyznacz n, wiedząc że n\in N_+
a)
n+{n\choose 2}=15
n+\frac{n!}{(n-2)!\cdot 2!}=15
n+\frac{(n-2)!\cdot (n-1)\cdot n}{(n-2)!\cdot 2} =15
n+\frac{(n-1)n}{2}=15 \ |*2
2n+n^2-n=30
n^2+n=30=0
a=1, b=1, c=-30
\Delta=b^2-4ac=1-4\cdot 1\cdot (-30)=1+120=121
\sqrt\Delta=11
n_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-1-11}{2\cdot 1}=-6\not\in N_+ odrzucamy
n_2=\frac{-b+\sqrt\Delta}{2a}=\frac{-1+11}{2\cdot 1}=\frac{10}{2}=5
n=5
b)
{n\choose n-1}+{2n\choose 2n-2}=18
\frac{n!}{[n-(n-1)]!\cdot (n-1)!}+\frac{(2n)!}{[2n-(2n-2)]!\cdot (2n-2)!}=18
\frac{(n-1)\cdot n}{1!\cdot (n-1)!}+\frac{(2n-2)!\cdot (2n-1)\cdot 2n}{2!\cdot (2n-2)!}=18
n+\frac{(2n-1)\cdot \not2^1n}{\not2^1}=18
n+(2n-1)n=18
n+2n^2-n=18
2n^2=18 \ |:2
n^2=9
n=3