Esame di Stato 2008 – 6

Se $\displaystyle {n \choose 1}$ , $\displaystyle {n \choose 2}$ , $\displaystyle {n \choose 3}$ , con n>3, sono in progressione aritmetica, qual è il valore di n?

$\displaystyle {n \choose k} =\frac{n!}{k!(n-k)!}$

$\displaystyle {n \choose k} =\frac{n(n-1)\ \cdots\ (n-k+1)}{k!}$

$\displaystyle n \ge k \ge 0$

$\displaystyle n\ge1$ , $\displaystyle n\ge2$ , $\displaystyle n\ge3$ ⇒ $\displaystyle n \ge 3$

Soluzione 1

$\displaystyle {n \choose 1}+x={n \choose 2}$

$\displaystyle {n \choose 2}+x={n \choose 3}$

Si tratta di risolvere l’equazione

$\displaystyle {n \choose 3}-{n \choose 2}={n \choose 2}-{n \choose 1}$

$\displaystyle {n \choose 3}-2{n \choose 2}+{n \choose 1}=0$

…

n=7

Soluzione 2

Calcola i coefficienti binomiali

n	$\displaystyle {n \choose 1}$	$\displaystyle {n \choose 2}$	$\displaystyle {n \choose 3}$
3	$\displaystyle {3 \choose 1}$ = 3	$\displaystyle {3 \choose 2}$ = 3	$\displaystyle {3 \choose 3}$ = 1
4	$\displaystyle {4 \choose 1}$ = 4	$\displaystyle {4 \choose 2}$ = 6	$\displaystyle {4 \choose 3}$ = 4
5	$\displaystyle {5 \choose 1}$ = 5	$\displaystyle {5 \choose 2}$ = 10	$\displaystyle {5 \choose 3}$ = 10
6	$\displaystyle {6 \choose 1}$ = 6	$\displaystyle {6 \choose 2}$ = 15	$\displaystyle {6 \choose 3}$ = 20
7	$\displaystyle {7 \choose 1}$ = 7	$\displaystyle {7 \choose 2}$ = 21	$\displaystyle {7 \choose 3}$ = 35

e individua la progressione aritmetica

7
21 (7+14)
35 (21+14)

Soluzione 3

Genera le combinazioni, contale e individua la progressione aritmetica

n	$\displaystyle {n \choose 1}$	$\displaystyle {n \choose 2}$	$\displaystyle {n \choose 3}$
3	a b c 3	ab ac bc 3	abc 1
4	a b c d 4	ab ac ad bc bd cd 6	abc abd acd bcd 4
5	a b c d e 5	ab ac ad ae bc bd be cd ce de 10	abc abd abe acd ace ade bcd bce bde cde 10
6	a b c d e f 6	ab ac ad ae af bc bd be bf cd ce cf de df ef 15	abc abd abe abf acd ace acf ade adf aef bcd bce bcf bde bdf bef cde cdf cef def 20
7	a b c d e f g 7	ab ac ad ae af ag bc bd be bf bg cd ce cf cg de df dg ef eg fg 21	abc abd abe abf abg acd ace acf acg ade adf adg aef aeg afg bcd bce bcf bcg bde bdf bdg bef beg bfg cde cdf cdg cef ceg cfg def deg dfg efg 35