a-b-c-1-2-3-n-find-a-b-c-abc-

Question Number 87313 by mr W last updated on 03/Apr/20

a, b, c = 1, 2, 3, \dots, n

f i n d \sum_{a \neq b \neq c} a b c

Commented by mr W last updated on 04/Apr/20

y e s, y o u a r e r i g h t .

Commented by mr W last updated on 04/Apr/20

f o r e x a m p l e n = 4

\sum_{a \neq b \neq c} a b c = 1 \times 2 \times 3 + 1 \times 2 \times 4 + 1 \times 3 \times 4 + 2 \times 3 \times 4 = 50

Commented by mr W last updated on 03/Apr/20

t o c h o o s e t h r e e d i f f e r e n t n u m b e r s

f r o m n n u m b e r s t h e r e a r e C_{3}^{n}

c o m b i n a t i o n s . f r o m e a c h c o m b i n a t i o n

w e g e t t h e p r o d u c t o f t h e t h r e e n u m b e r s .

n o w i t i s t o f i n d t h e s u m o f a l l t h e s e

p r o d u c t s .

Commented by MJS last updated on 04/Apr/20

sorry but for n = 4 we have

1 \times 2 \times 3 + 1 \times 2 \times 4 + 1 \times 3 \times 4 + 2 \times 3 \times 4 = 50

Answered by mr W last updated on 04/Apr/20

S = Σ a b c = (\underset{a = 1}{\sum^{n}} a) (\underset{b = 1}{\sum^{n}} b) (\underset{c = 1}{\sum^{n}} c) = {[\frac{n (n + 1)}{2}]}^{3}

S_{1} = \sum_{a = b = c} a b c = \underset{a = 1}{\sum^{n}} a^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}

S_{2} = \sum_{a, b = c \neq a} a b c = 3 \underset{b = 1}{\sum^{n}} [(\underset{a = 1}{\sum^{n}} a - b) (b^{2})]

= 3 \underset{b = 1}{\sum^{n}} [(\underset{a = 1}{\sum^{n}} a) b^{2} - b^{3}]

= 3 [(\underset{a = 1}{\sum^{n}} a) (\underset{b = 1}{\sum^{n}} b^{2}) - (\underset{b = 1}{\sum^{n}} b^{3})]

= 3 [\frac{n (n + 1)}{2} \times \frac{n (n + 1) (2 n + 1)}{6} - \frac{n^{2} {(n + 1)}^{2}}{4}]

= \frac{(n - 1) n^{2} {(n + 1)}^{2}}{2}

S_{3} = \sum_{a \neq b \neq c} a b c = \frac{1}{3!} (S - S_{1} - S_{2})

= \frac{1}{6} [\frac{n^{3} {(n + 1)}^{3}}{8} - \frac{n^{2} {(n + 1)}^{2}}{4} - \frac{(n - 1) n^{2} {(n + 1)}^{2}}{2}]

= \frac{(n - 2) (n - 1) n^{2} {(n + 1)}^{2}}{48}

e x a m p l e : n = 4

S_{3} = \frac{2 \times 3 \times 4^{2} \times 5^{2}}{48} = 50

1 \times 2 \times 3 + 1 \times 2 \times 4 + 1 \times 3 \times 4 + 2 \times 3 \times 3

= 50

e x a m p l e : n = 5

S_{3} = \frac{3 \times 4 \times 5^{2} \times 6^{2}}{48} = 225

1 \times 2 \times 3 + 1 \times 2 \times 4 + 1 \times 2 \times 5 + 1 \times 3 \times 4 + 1 \times 3 \times 5 + 1 \times 4 \times 5

+ 2 \times 3 \times 4 + 2 \times 3 \times 5

+ 3 \times 4 \times 5

= 225

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