The-sum-of-all-possible-products-taken-two-at-a-time-out-of-the-numbers-1-2-3-4-is-a-0-b-30-c-30-d-55-

Question Number 203788 by BaliramKumar last updated on 28/Jan/24

The sum of all possible products taken two at

a time out of the numbers \pm 1, \pm 2, \pm 3, \pm 4 is

(a) 0 (b) - 30 (c) 30 (d) 55

Answered by mr W last updated on 28/Jan/24

t o t a k e t w o f r o m 8 t h e r e a r e

C_{8}^{2} = \frac{8 \times 7}{2} = 28 p o s s i b i l i t i e s

g r o u p 1 : (+ 1, + 2, + 3, + 4)

g r o u p 2 : (- 1, - 2, - 3, - 4)

w h e n w e t a k e t w o n u m b e r s, t h e r e

a r e f o l l o w i n g c a s e s :

c a s e 1 :

b o t h n u m b e r s f r o m g r o u p 1

C_{4}^{2} = \frac{4 \times 3}{2} = 6 p o s s i b i l i t i e s

1 \times 2 + 1 \times 3 + 1 \times 4 + 2 \times 3 + 2 \times 4 + 3 \times 4 = 35

c a s e 2 :

b o t h n u m b e r s f r o m g r o u p 2

a l s o 6 p o s s i b i l i t i e s

s u m i s a l s o 35 .

c a s e 3 :

o n e n u m b e r f r o m g r o u p 1 a n d t h e

o t h e r n u m b e r f r o m g r o u p 2

4 \times 4 = 16 p o s s i b i l i t i e s

(- 1 - 2 - 3 - 4) \times (1 + 2 + 3 + 4) = - 100

s u m m a r y :

6 + 6 + 16 = 28 p o s s i b i l i t i e s ✓

s u m = 35 + 35 - 100 = - 30 ✓

\Rightarrow a n s w e r (b) i s c o r r e c t

Commented by mr W last updated on 28/Jan/24

i t h i n k w e c a n g e n e r a l i z e :

{l e t}^{'} s s a y w e t a k e t w o n u m b e r s o u t

o f \pm 1, \pm 2, \pm 3, \dots, \pm n

t h e s u m o f a l l p o s s i b l e p r o d u c t s i s

- (1^{2} + 2^{2} + 3^{2} + \dots + n^{2}) = - \frac{n (n + 1) (2 n + 1)}{6}

e . g . n = 4 : - \frac{4 \times 5 \times 9}{6} = - 30 ✓

Commented by BaliramKumar last updated on 28/Jan/24

Thanks Sir