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We have a 2x2 grid and 10 different colours. I want to paint such that adjacent grids are painted with different colors. How many ways can i do this?

I labeled the grids as:
A B
C D

Scenario 1: B and C different colors.
1)A has 10 colors to choose from.
2)B has 9 colors to choose from
3)C has 8 colors to choose from
4)D has 8 colors to choose from
Total ways: 10*9*8*8 = 5760 ways

Scenario 2: B and C have same colors.
1)A has 10 colors to choose from.
2)B has 9 colors to choose from
3)C has 9 colors to choose from
4)D has 9 colors to choose from
Total ways: 10*9*9*9 = 7290 ways

Total ways = 7290 + 5760 = 13050 ways.

Did i do it correctly?
joriki
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1 Answers1

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As Jennifer pointed out in a comment, you have one excess factor $9$ in scenario $2$. Correcting that yields $10\cdot9\cdot8\cdot8+\cdot10\cdot9\cdot9=90(64+9)=6570$.

Another way to do this is via inclusion-exclusion. We have four conditions, one for each pair of adjacent cells. There are $10^{4-k}$ options that fulfil $k$ conditions, except for $k=4$ there are $10$ options instead of $1$, for a total of

$$ 9+\sum_{k=0}^4(-1)^k10^{4-k}\binom4k=9+10^4\left(1-\frac1{10}\right)^4=9+9^4=6570. $$

Another solution: Start at A with $10$ options and go clockwise, with $9$ options per step, which makes $10\cdot9^3$. Then subtract the configurations in which A and C have the same colour, of which there are $10\cdot9\cdot8$, for a total of $10\cdot9(81-8)=6570$.

If you just want a rough estimate, start at A and go clockwise; by the time you get back to A the information which colour you started with will almost be lost, so the probability that you end up with the same colour is almost exactly $1$ in $10$, which effectively reduces the options for the first colour from $10$ to $9$, yielding an estimate of $9^4=6561$.

See also In how many ways can we colour $n$ baskets with $r$ colours?

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joriki
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  • I like using awk to verify my results for simple combinatorics. Here, awk -v n=10 'BEGIN { for (a=0;a<n;a++) for (b=0;b<n;b++) for (c=0;c<n;c++) for (d=0;d<n;d++) i += (a!=b && a!=c && b!=d && c!=d); printf "%d\n", i }' will print 6570 as one would expect. For n=100, it takes almost half a minute to reach the result (96059700), so for brute-forcing large problems it's not that good. It's a simple language, but has associative arrays, which helps with many types of combinatorics problems. – Nominal Animal Apr 23 '16 at 09:57