A coin’s faces are marked with 0 and 1. What is the probability of scoring K 1s in N tosses?

Let’s start with 3 tosses (N=3) and we want to find out the probability of scoring 2 (K=2) 1s. Let’s map out the possible outcomes of three coin tosses.

N | Possible Outcomes | Possibility of scoring K 1s |

2 | 00 01 10 11 | 0: 1 1: 2 2: 1 |

3 | 000 001 010 011 100 101 110 111 | 0: 1 1: 3 2: 3 3: 1 |

4 | 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 | 0: 1 1: 4 2: 6 3: 4 4: 1 |

And if you noticed, that exactly looks like Pascal’s triangle! Just get the Kth entry from the Pascal’s triangle’s Nth row.