The-total-number-of-numbers-of-not-more-than-20-digits-that-are-formed-by-using-the-digits-0-1-2-3-and-4-is-

Question Number 43787 by kkr1729 last updated on 15/Sep/18

The total number of numbers of not

more than 20 digits that are formed

by using the digits 0, 1, 2, 3 and 4 is

Answered by MrW3 last updated on 17/Sep/18

l e t n u m b e r o f n u m b e r s w i t h n d i g i t s = S (n)

S (1) = 4

S (2) = 4 \times 5 = 4 \times 5^{1}

S (3) = 4 \times 5 \times 5 = 4 \times 5^{2}

S (4) = 4 \times 5 \times 5 \times 5 = 4 \times 5^{3}

S (n) = 4 \times 5^{n - 1}

S (20) = 4 \times 5^{19}

\underset{n = 1}{\sum^{20}} S (n) = \frac{4 \times (5^{20} - 1)}{5 - 1} = 5^{20} - 1

I f n o t 5 d i g i t s (0, 1, 2 \dots 4) b u t 10 d i g i t s

(0, 1, 2 \dots . 9) a r e u s e d, w e k n o w t h e r e a r e

10^{20} - 1 n u m b e r s w i t h m a x . 20 d i g i t s :

1, 2, 3, \dots, \underset{20 t i m e s 9}{9999 \dots . 9}

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