## 1180.45 – Biology Lesson

In every cup of yoghurt there are, let us say, over a million lacto-bacteria cells. Each of these cells has between one and 10 cilia. Each cilium has at between one and 10 organelles. Under these (hypothetical) circumstances, which of the statements below is definitely true, which are definitely false, and which might be true or false.

1. There are thousands of lacto-bacteria cells in a cup of yoghurt with the same number of cilia.

2. There are thousands of lacto-bacteria cells in a cup of yoghurt with the same total number of organelles on their cilia.

3. There are thousands of lacto-bacteria cells in a cup of yoghurt all of whom have the same number of cilia and all of whose cilia have the same number of organelles.

4. There are thousands of lacto-bacteria in a cup of yoghurt with exactly five cilia.

Solution

Remember, there are over 1,000,000 lacto-bacteria cells in your cup of yoghurt. Each one has 1 to 10 cilia. Each cilia has 1 to 10 organelles. Now we must determine the truth or falsity of the four given statements. We will take them one at a time.

1. Now then: "There are thousands of lacto-bacteria cells with the same number of cilia." There are 1,000,000 bacteria cells but only 10 different numbers of cilia. Even if we spread the cells out like dealing a deck of 1,000,000 cards into 10 hands, roughly a tenth or 100,000 (one hundred thousand) of the cells will have the same number of celia. So this statement is true.

2. "There are thousands of cilia with the same number of organelles."

Likewise, this is true. There are at least 1,000,000 cilia (because each bacteria has at least one cilium) and as with statement 1 we are dividing them into only 10 different kinds (1 to 10 organelles). At least a thousand, in fact at least 100,000, will have the same number of organelles. So this statement is true.

1. "There are thousands of lacto-bacteria whose cilia have the same numbers of organelles."

This is trickier. How many different kinds of bacteria have just one cilium? The answer is 10 because that one cilium will have 1 to 10 organelles. How many bacteria have 2 cilia? Well the first cilium can be one of ten types and the second cilium also is one of ten types. That makes 100 = 102 possible combinations of types for the two cilia. Now if the bacterium has three cilia, then these can come in 103 different arrangements. What we conclude is that if the bacterium can have 1 or 2 or 3 or . . . up to 10 cilia, then the bacterium can be of

10 + 10^2 + 10^3 + 10^4 + 10^5 + 10^6 + 10^7 + 10^8 + 10^9 + 10^10 = 11,111,111,110

different kinds, depending on the number of cilia and the number of organelles on each cilium. This number is more than eleven times the total number of bacteria in the cup. So NO. The bacteria, all one million or more of them, might all have completely different arrangements of cilia. Maybe it is unlikely, but this statement is possibly false although it also might be true, if, for example, all the bacteria were identical in their cilia.

1. Thousands of lacto-bacteria have exactly five cilia.

This is like the previous statement. It is possible that thousands of lacto-bacteria have just 5 cilia but not likely. This is possibly true and possibly false.