n^k

10 * 9 * 8 * 7 * 6 * 5 * 4

10 * (10 - 1) * (10 - 2) * (10 - 3) * (10 - 4) * (10 - 5) * (10 - 6)

President | Vice-President | Treasurer | Secretary |
---|---|---|---|

Jack | Jill | Johnny | Mary |

Jill | Jack | Johnny | Mary |

Mary | Jack | Jill | Johnny |

Mary | Jill | Johnny | Jack |

n*(n-1)(n-2)\dots(n-k+1)

= \frac{n!}{(n-k)!}

=\frac{n*(n-1)(n-2)\dots(n-k+1)(n-k)(n-k-1)(n-k-2)\dots3*2*1}{(n-k)(n-k-1)(n-k-2)\dots3*2*1}

count(B) = count(A) - count(C)

count(A) = 26*25*24

count(C) = 21*20*19

count(B) = 26*25*24 - 21*20*19

count(A) = 26^5

count(C) = 26*25^4

count(C) = 26^5 - 26*25^4

\neq

\frac{n!}{(n-k)!}

= N * k!

= \frac{n!}{(n-k)!}

\therefore N = \frac{n!}{(n-k)!k!}

Collections | Sequences |
---|---|

(a,e,i) | {(a,e,i), (a,i,e), (e,a,i), (e,i,a), (i,a,e),(i,e,a)} |

(a,e,o) | {(a,e,o), (a,o,e), (e,a,o), (e,o,a), (o,a,e), (o,e,a)} |

(a,e,u) | {(a,e,u), (a,u,e), (e,a,u), (e,u,a), (u,a,e), (u,e,a)} |

(a,i,o) | {(a,i,o), (a,o,i), (i,a,o), (i,o,a), (o,a,i), (o,i,a)} |

(a,i,u) | {(a,i,u), (a,u,i), (i,a,u), (i,u,a), (u,a,i), (u,i,a)} |

(a,o,u) | {(a,o,u), (a,u,o), (o,a,u), (o,u,a), (u,a,o), (u,o,a)} |

(e,i,o) | {(e,i,o), (e,o,i), (i,e,o), (i,o,e), (o,e,i), (o,i,e)} |

(e,i,u) | {(e,i,u), (e,u,i), (i,e,u), (i,u,e), (u,e,i), (u,i,e)} |

(e,o,u) | {(e,o,u), (e,u,o), (o,e,u), (o,u,e), (u,e,o), (u,o,e)} |

(i,o,u) | {(i,o,u), (i,u,o), (o,i,u), (o,u,i), (u,i,o), (u,o,i)} |

\frac{5!}{~2!~3!~}

Sequences |
---|

{(a,e,i), (a,i,e), (e,a,i), (e,i,a), (i,a,e),(i,e,a)} |

{(a,e,o), (a,o,e), (e,a,o), (e,o,a), (o,a,e), (o,e,a)} |

{(a,e,u), (a,u,e), (e,a,u), (e,u,a), (u,a,e), (u,e,a)} |

{(a,i,o), (a,o,i), (i,a,o), (i,o,a), (o,a,i), (o,i,a)} |

{(a,i,u), (a,u,i), (i,a,u), (i,u,a), (u,a,i), (u,i,a)} |

{(a,o,u), (a,u,o), (o,a,u), (o,u,a), (u,a,o), (u,o,a)} |

{(e,i,o), (e,o,i), (i,e,o), (i,o,e), (o,e,i), (o,i,e)} |

{(e,i,u), (e,u,i), (i,e,u), (i,u,e), (u,e,i), (u,i,e)} |

{(e,o,u), (e,u,o), (o,e,u), (o,u,e), (u,e,o), (u,o,e)} |

{(i,o,u), (i,u,o), (o,i,u), (o,u,i), (u,i,o), (u,o,i)} |

ABCD |
---|

ABDC |

ACBD |

ACDB |

ADBC |

ADCB |

BACD |

BADC |

BCAD |

BCDA |

BDAC |

BDCA |

CABD |

CADB |

CBAD |

CBDA |

CDAB |

CDBA |

DABC |
---|

DACB |

DBAC |

DBCA |

DCAB |

DCBA |

ABCD |
---|

ABDC |

ACBD |

ACDB |

ADBC |

ADCB |

BACD |

BADC |

BCAD |

BCDA |

BDAC |

BDCA |

CABD |

CADB |

CBAD |

CBDA |

CDAB |

CDBA |

DABC |
---|

DACB |

DBAC |

DBCA |

DCAB |

DCBA |

\frac{15!}{~~~~~(15-4)!~~~~~~}

\frac{~}{4!}

n \choose k

\frac{n!}{(n-k)!k!}

{10 \choose 2} = 45

{10 \choose 3} = 120

{6 \choose 2} = 15

{n \choose 3}

\frac{n!}{(n-k)!}

n^k

\frac{n!}{(n-k)!k!}

?

{14 \choose 5}

{n+k-1 \choose k}

{n+k-1 \choose k}

\frac{n!}{(n-k)!}

n^k

\frac{n!}{(n-k)!k!}

{n+k-1 \choose k}

{8 \choose 2}

{7 \choose 2}

*

{7 \choose 5}

{2 \choose 1}

{4 \choose 3}

{3 \choose 2}

{7 \choose 5}

{2 \choose 1}

{4 \choose 3}

{3 \choose 2}

*

*

*

m_1 + m_2 + \dots + m_i = n

k_1 + k_2 + \dots + k_i = k

Given:n~items~of~i~different~types

{m_1 \choose k_1} * {m_2 \choose k_2} * \cdots * {m_i \choose k_i}

Form:collection~of~k~items

count(A) = {21 \choose 4}

count(C) = {16 \choose 4}

count(B) = {21 \choose 4} - {16 \choose 4}

\frac{n!}{(n-k)!}

n^k

\frac{n!}{(n-k)!k!}

{n+k-1 \choose k}