Python study notes: permutation and combination

1. Permutation and combination formula

1. Arrangement formula

$P_n^m=n(n-1)(n-2)...(n-m+1)=\frac{n!}{(n-m)!}$
$P_n^n=n!(0!=1)$

2. Combination formula

$C_n^m=\frac{n(n-1)(n-2)...(n-m+1)}{m!}=\frac{n!}{(n-m)!m!}$

Two, calculate permutation and combination

1. The task of placing books

• Ms. Jones wants to put $.\; 10$ book is placed on the shelf, which has$4$ math books,$3$ chemistry books,$2$ history books and$1$ language document. Now Ms. Jones wants to organize her books. If the same books must be put together, how many ways are there?
• If you put the math book first, then the chemistry book, then the history book, and finally the language book, then there is $P_4^4\times P_3^3\times P_2^2\times P_1^1=4!\times 3!\times 2!\times 1!$ Ways, and these4 4The order of the$4$ disciplines is$P_4^4=4!$ Types, so the answer is$4!\times 4!\times 3!\times 2!\times 1!=6912$
• Use the perm() function in the scipy.special module to calculate
  

2. Committee composition

• A group has $12$ people, of which$5$ ladies,$7$ men, now select$2$ ladies and$Three$ men form a committee. How many different committees are there?
• Because there is $C_5^2$There are two ways to select women, there are $C_7^3$There are two ways to select men, so according to the basic counting rule, one is $C_5^2\times C_7^3=\frac{5\times 4}{2\times 1}\times \frac{7\times 6\times 5}{3\times 2\times 1}=350$ possible Commission.
• Use the comb() function in the scipy.special module to calculate
  

Three, display permutation and combination

1. Show full arrangement

• 1, 2, 3 all arrangements

2. Display arrangement

• Arrangement of 3 of 1, 2, 3, 4, and 5
  

3. Display combination

• Take 3 of 1, 2, 3, 4, and 5 to form a group. How many possible groups are there in total?