Think inside the box!

Number Theory Level 3

You have a big box containing 10 boxes inside each of them 9 boxes and each of the 9 boxes contains 8 boxes and so on until you end up with empty boxes. What is the total number of boxes?

3628800 9864101 7257601 6840101 36288000

Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

Ossama Ismail
Mar 29, 2018

ِAns: 9864101

Total number of boxes can be found by finding the number of all types of boxes as follows:-

1 − 1 = 1 Big box 10 − 10 ∗ 1 = 10 9 − 10 ∗ 9 = 90 8 − 10 ∗ 9 ∗ 8 = 720 7 − 10 ∗ 9 ∗ 8 ∗ 7 = 5040 6 − 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 = 30240 5 − 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 = 151200 4 − 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 = 604800 3 − 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 = 1814400 2 − 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 = 3628800 1 − 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 = 3628800 Empty boxes Total = 9864101 \begin{aligned} 1 - &1 &= 1 &\text{ Big box}&\\ 10 - &10*1 &=10 &\\ 9 - &10*9 &= 90&\\ 8- & 10*9*8 &= 720&\\ 7- & 10*9*8*7 &= 5040&\\ 6- & 10*9*8*7*6 &= 30240&\\ 5- & 10*9*8*7*6*5 &= 151200&\\ 4- & 10*9*8*7*6*5*4 &= 604800& \\ 3- & 10*9*8*7*6*5*4*3 &=1814400&\\ 2- &10*9*8*7*6*5*4 *3*2 &= 3628800&\\ 1- & 10*9*8*7*6*5*4 *3*2 &= 3628800& \ \ \text{Empty boxes}\\ & \large \text{ Total} &= 9864101&\\ \end{aligned}

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