$\chi^2$ Test of Independence [duplicate]

I am trying to conduct a $$\chi^2$$ test of independence based on the following data.

          Type 1 Type 2  Type 3
House 1       36      1       2
House 2       32      6       2
House 3        2      0       1
House 4        6      0       0
House 5        1      0       2


I understand how to complete such a question, however I encounter a problem calculating the expected frequencies. That is, a lot of the values are less than 5.

              Type 1      Type 2      Type 3
House 1    32.088889   2.9555556   2.9555556
House 2    33.777778   3.1111111   3.1111111
House 3     2.533333   0.2333333   0.2333333
House 4     5.066667   0.4666667   0.4666667
House 5     2.533333   0.2333333   0.2333333


I have seen in One-Way Frequency tables where one category is collapsed into another if one of the expected frequencies is less than 5. However, I have not encountered such a technique in a Two-Way test of independence, especially when there are so many expected frequencies that are less than 5.

Any suggestions on how to deal with this situation without using Fisher's Exact Test?

• Could you explain what you mean by "not permitted"? What precludes the Fisher Test?
– whuber
Commented Jan 18, 2021 at 13:45
• What I mean is, can this question be completed without using Fisher's Exact Test? Commented Jan 18, 2021 at 17:36
• Commented Apr 7, 2022 at 22:40