Esame di Stato 2007 – 8

di

Si risolva l’equazione \displaystyle 4{n \choose 4} = 15{n-2 \choose 3}

Osserva

  • n \ge 4
  • n-2 \ge 3

quindi: n \ge 5

Risolvi l’equazione

  • \displaystyle 4\ {n \choose 4} = 15\ {n-2 \choose 3}
  • \displaystyle 4\ \frac{n!}{4!(n-4)!} = 15\ \frac{(n-2)!}{3!(n-2-3)!}
  • \displaystyle \frac{4\cdot n(n-1)(n-2)(n-3)(n-4)(n-5)!}{4\cdot 3!\cdot (n-4)(n-5)!} – \displaystyle \frac{15\cdot (n-2)(n-3)(n-4)(n-5)!}{3!\cdot (n-5)!} = 0
  • …
  • n_1 = 6
    n_2 = 10

Esercizio

Prova per n= 5, 6, … finché non trovi le soluzioni

n\displaystyle 4\ {n \choose 4}\displaystyle 15\ {{n-2} \choose 3}
5\displaystyle 4\ {5 \choose 4}= 4 * 5= 20\displaystyle 15\ {{3} \choose 3}= 15 * 1= 15
6\displaystyle 4\ {6 \choose 4}= 4 * 15= 60\displaystyle 15\ {{4} \choose 3}= 15 * 4= 60
7\displaystyle 4\ {7 \choose 4}= 4 * 35= 140\displaystyle 15\ {{5} \choose 3}= 15 * 10= 150
8\displaystyle 4\ {8 \choose 4}= 4 * 70= 280\displaystyle 15\ {{6} \choose 3}= 15 * 20= 300
9\displaystyle 4\ {9 \choose 4}= 4 * 126= 504\displaystyle 15\ {{7} \choose 3}= 15 * 35= 525
10\displaystyle 4\ {10 \choose 4}= 4 * 210= 840\displaystyle 15\ {{8} \choose 3}= 15 * 56= 840
11\displaystyle 4\ {11 \choose 4}= 4 * …= …\displaystyle 15\ {{9} \choose 3}= 15 * …= …