Răspuns:
2) a) -3
b) 0
c) a = b = 8
Explicație pas cu pas:
2)
[tex]C_{n}^{k}[/tex] = [tex]\frac{n!}{k! (n-k)!}[/tex]
[tex]A_{n}^{k}[/tex] = [tex]\frac{n!}{(n-k)!}[/tex]
[tex]P_{3}[/tex] = n!·
a) [tex]C_{3}^{2}[/tex] - [tex]A_{4}^{2}[/tex] + [tex]P_{3}[/tex]
[tex]C_{3}^{2}[/tex] = [tex]\frac{3!}{2! (3-2)!}[/tex] = [tex]\frac{3!}{2! (1)!}[/tex] = [tex]\frac{3!}{2! }[/tex] = [tex]\frac{1X2X3}{1X2}[/tex] = 3
[tex]A_{4}^{2}[/tex] = [tex]\frac{4!}{(4-2)!}[/tex] = [tex]\frac{4!}{(2)!}[/tex] = [tex]\frac{1X2X3X4}{1X2}[/tex] = 3 x 4 = 12 (unde X este semnul pentru înmulțire)
[tex]P_{3}[/tex] = 3! = 1 · 2 · 3 = 6
[tex]C_{3}^{2}[/tex] - [tex]A_{4}^{2}[/tex] + [tex]P_{3}[/tex] = 3 - 12 + 6 = - 3
b) [tex]C_{4}^{0}[/tex] - [tex]C_{4}^{1}[/tex] + [tex]C_{4}^{2}[/tex] - [tex]C_{4}^{3}[/tex] + [tex]C_{4}^{4}[/tex] =
[tex]C_{4}^{0}[/tex] = [tex]\frac{4!}{0! (4-0)!}[/tex] = [tex]\frac{4!}{0! (4)!}[/tex] = [tex]\frac{4!}{(4)!}[/tex] = 1
[tex]C_{4}^{1}[/tex] = [tex]\frac{4!}{1! (4-1)!}[/tex] = [tex]\frac{4!}{1! (3)!}[/tex] = [tex]\frac{4!}{(3)!}[/tex] =[tex]\frac{1X2X3X4}{1X2X3}[/tex] = 4
[tex]C_{4}^{2}[/tex] = [tex]\frac{4!}{2! (4-2)!}[/tex] = [tex]\frac{4!}{2! (2)!}[/tex] = [tex]\frac{4!}{(2)!}[/tex] = [tex]\frac{1X2X3X4}{1X2X1X2}[/tex] = (3 x 4)/2 = 12/2 = 6
[tex]C_{4}^{3}[/tex] = [tex]\frac{4!}{3! (4-3)!}[/tex] = [tex]\frac{4!}{3! (1)!}[/tex] = [tex]\frac{4!}{(3)!}[/tex] = [tex]\frac{1X2X3X4}{1X2X3}[/tex] = 4
[tex]C_{4}^{4}[/tex] = [tex]\frac{4!}{4! (4-4)!}[/tex] = [tex]\frac{4!}{4! (0)!}[/tex] = [tex]\frac{4!}{(4)!}[/tex] = 1
[tex]C_{4}^{0}[/tex] - [tex]C_{4}^{1}[/tex] + [tex]C_{4}^{2}[/tex] - [tex]C_{4}^{3}[/tex] + [tex]C_{4}^{4}[/tex] = 1 - 4 + 6 - 4 + 1 = 0
b) a = [tex]C_{4}^{1}[/tex] + [tex]C_{4}^{3}[/tex]
de la exercițiul anterior știm că
[tex]C_{4}^{1}[/tex] = [tex]\frac{4!}{1! (4-1)!}[/tex] = [tex]\frac{4!}{1! (3)!}[/tex] = [tex]\frac{4!}{(3)!}[/tex] =[tex]\frac{1X2X3X4}{1X2X3}[/tex] = 4
[tex]C_{4}^{3}[/tex] = [tex]\frac{4!}{3! (4-3)!}[/tex] = [tex]\frac{4!}{3! (1)!}[/tex] = [tex]\frac{4!}{(3)!}[/tex] = [tex]\frac{1X2X3X4}{1X2X3}[/tex] = 4
deci a = [tex]C_{4}^{1}[/tex] + [tex]C_{4}^{3}[/tex] = 4 + 4 = 8
b = [tex]C_{3}^{0}[/tex] + [tex]C_{3}^{1}[/tex] + [tex]C_{3}^{2}[/tex] + [tex]C_{3}^{3}[/tex]
[tex]C_{3}^{0}[/tex] = [tex]\frac{3!}{0! (3-0)!}[/tex] = [tex]\frac{3!}{0! (3)!}[/tex] = [tex]\frac{3!}{(3)!}[/tex] = 1
[tex]C_{3}^{1}[/tex] = [tex]\frac{3!}{1! (3-1)!}[/tex] = [tex]\frac{3!}{1! (2)!}[/tex] = [tex]\frac{3!}{(2)!}[/tex] =[tex]\frac{1X2X3}{1X2}[/tex] = 3
[tex]C_{3}^{2}[/tex] = [tex]\frac{3!}{2! (3-2)!}[/tex] = [tex]\frac{3!}{2! (1)!}[/tex] = [tex]\frac{3!}{(2)!}[/tex] = [tex]\frac{1X2X3}{1X2}[/tex] = 3
[tex]C_{3}^{3}[/tex] = [tex]\frac{3!}{3! (3-3)!}[/tex] = [tex]\frac{3!}{3! (0)!}[/tex] = [tex]\frac{3!}{(3)!}[/tex] = 1
b = [tex]C_{3}^{0}[/tex] + [tex]C_{3}^{1}[/tex] + [tex]C_{3}^{2}[/tex] + [tex]C_{3}^{3}[/tex] = 1 + 3 + 3 + 1 = 8
deci a = b