[Tutorial] Product Trick

Revision en8, by TheScrasse, 2023-06-25 14:59:37

Hello everyone,
in this tutorial we will see a trick that can be useful in combinatorics and/or DP tasks. In particular, you can use it when the statement says something similar to "the score of an array is the product of its elements, find the sum of the scores over all the possible arrays".

Prerequisites: basic combinatorics and DP

## The trick

The trick is very simple.

"The score of an array $a$ is $\prod_{i=1}^n a_i$" can be rephrased as "if there are $n$ boxes, and the $i$-th box contains $a_i$ distinguishable balls, the score of $a$ is equal to the number of ways to color a ball for each box".

This is quite obvious, but it can be extremely powerful. Let's see some problems that are trivialized by this trick.

## Dwango Programming Contest 6th, problem C (rating: 2618)

Hint 1
Hint 2
Hint 3
Hint 4
Hint 5
Solution

Implementation (C++)

## abc231_g (rating: 2606)

Hint 1
Hint 2
Hint 3
Hint 4
Hint 5
Solution
Bonus

Implementation (C++)

## arc124_e (rating: 3031)

Hint 1
Hint 2
Hint 3
Hint 4
Solution

Implementation (C++)

## Other problems

arc147_d - Sets Scores (rating: 2145)
abc214_g - Three Permutations (rating: 2893) (suggested by __hermit__)
agc013_e - Placing Squares (rating: 3455) (zscoder)
1842G - Tenzing and Random Operations
IOI 2022/4 - Digital Circuit
abc225_h - Social Distance 2 (rating: 3061) (__hermit__)

## Conclusions

We've seen that "product trick" is very useful to find a DP that solves the problem. There exist similar counting tricks: for example, "The score of an array $a$ is $\sum_{i=1}^n a_i^2$" can be rephrased as "if there are $n$ boxes, and the $i$-th box contains $a_i$ distinguishable balls, the score of $a$ is equal to the number of ordered pairs of balls belonging to the same box" (you can try to use it in 1278F - Cards).

Of course, suggestions/corrections are welcome. In particular, please share in the comments other problems where you can use this trick.

I hope you enjoyed the blog!

#### History

Revisions

Rev. Lang. By When Δ Comment
en8 TheScrasse 2023-06-25 14:59:37 21 Tiny change: '5])<br>\n[IOI' -> '5])<br>\n[problem:1842G]<br>\n[IOI'
en7 TheScrasse 2022-09-05 08:19:42 129
en6 TheScrasse 2022-09-04 19:47:33 255 Added two problems
en5 TheScrasse 2022-01-03 02:00:00 2
en4 TheScrasse 2022-01-03 01:28:59 230 Tiny change: 'r:Hermit])\n[abc225_' -> 'r:Hermit])<br>\n[abc225_'
en3 TheScrasse 2022-01-02 22:32:29 2 Tiny change: '{a_i^2(a_i-1)}{2} - \' -> '{a_i^2(a_i+1)}{2} - \'
en2 TheScrasse 2022-01-02 21:42:13 16
en1 TheScrasse 2022-01-02 21:30:58 8962 Initial revision (published)