The table below shows the possible results of drawing four marbles from a bag containing 50 green and 50 red marbles. Prepare a similar table for the following occurrence: You toss four coins into the air simultaneously and record all the possible results of the toss in terms of the numbers of heads and tails that can result. (For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved.) Possible Results of Drawing Four Marbles from a Bag End Result Possible Draws Total Number of Same Results All R RRRR 1 1G, 3R RRRG, RRGR, RGRR, GRRR 4 2G, 2R RRGG, RGRG, GRRG, RGGR, GRGR, GGRR 6 3G, 1R GGGR, GGRG, GRGG, RGGG 4 All G GGGG 1 (a) On the basis of your table, what is the most probable result of a toss? 3 heads and 1 tail 2 heads and 2 tails 3 tails and 1 head all heads or all tails (b) In terms of entropy, what is the most ordered state? 3 heads and 1 tail 2 heads and 2 tails 3 tails and 1 head all heads or all tails (c) In terms of entropy, what is the most disordered? 3 heads and 1 tail 2 heads and 2 tails 3 tails and 1 head all heads or all tails
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
The table below shows the possible results of drawing four marbles from a bag containing 50 green and 50 red marbles. Prepare a similar table for the following occurrence: You toss four coins into the air simultaneously and record all the possible results of the toss in terms of the numbers of heads and tails that can result. (For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved.)
| Possible Results of Drawing Four Marbles from a Bag | ||
|---|---|---|
| End Result | Possible Draws | Total Number of Same Results |
| All R | RRRR | 1 |
| 1G, 3R | RRRG, RRGR, RGRR, GRRR | 4 |
| 2G, 2R | RRGG, RGRG, GRRG, RGGR, GRGR, GGRR | 6 |
| 3G, 1R | GGGR, GGRG, GRGG, RGGG | 4 |
| All G | GGGG | 1 |
(b) In terms of entropy, what is the most ordered state?
(c) In terms of entropy, what is the most disordered?
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