I don\'t understand, where does the 7 come from for the case that if we allow ga

Question

I don't understand, where does the 7 come from for the case that if we allow gaps.

Example 1.2.5. (Master Mind) In the game of Master Mind, there are four gaps in a row, which can be filled with little coloured sticks. There are six different colours for the sticks. One can ask how many combinations there are to do this. This depends on the rules that we agree on If we insist that all four gaps are actually filled with different colours, then the number of ways to do this, is the same as the number of ways to choose an ordered set of size 4 from a set of six elements, that is, 6 × 5 × 4 × 3 360 If we allow gaps, that is, if we allow a gap not to be filled, then the number of possibilities is 7 × 6 × 5 × 4 = 840 Finally, if we allow gaps and the same colour to be used more than once, then the number of possibilities is just the number of ordered sequences of lengt.h 4 using seven symbols, that is, 74 2401. This last number appears in commercials for Master Mind as the total number of possible combinations

Explanation / Answer

If we allow gaps, but insist on filling them with different colours, then the first empty gap can either be filled with any of the 6 colours(6 possibilites) or we don't fill it(1 possibility). Thus for the first gap there are 7 possibilities (6 colours + 1 of leaving it unfilled) for the second gap there are 6 possibilites, for the third gap there are 5 possibilites and lastly for the fourth gap there are 4 possibilites.

Thus total no. of combinations = 7*6*5*4

Finally, if we allow gaps and same colour can be used more than once than each gap can be filled with any of the 7 possibilites. Thus total no. of combinations = 7*7*7*7 = 7^4

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