3n+1_conjecture 12, combination コンビネーション

投稿者: | 01/22/2022

Everyone loves it, it’s a combination. (nCk = n! / (n – k)! k!)
みんな大好き、コンビネーションです。

11C1 = 11 : odd -> (11 – 1 + 1) / 1 = 11 : odd
11C2 = 55 : odd -> (11 – 2 + 1) / 2 = 5 : odd
11C3 = 165 : odd -> (11 – 3 + 1) / 3 = 3 : odd
11C4 = 330 : even -> (11 – 4 + 1) / 4 = 2 : even
11C5 = 462 : even -> (11 – 5 + 1) / 5 = 7/5 : rational number

9C1 = 9 : odd -> (9 – 1 + 1) / 1 = 9 : odd
9C2 = 36 : even -> (9 – 2 + 1) / 2 = 4 : even
9C3 = 84 : even -> (9 – 3 + 1) / 3 = 7/3 : rational number

9C4 = 126 : even -> (9 – 4 + 1) / 4 = 3/2 : rational number

8C1 = 8 : even -> (8 – 1 + 1) / 1 = 8 : even
8C2 = 28 : even -> (8 – 2 + 1) / 2 = 7/2 : rational number
8C3 = 56 : even -> (8 – 3 + 1) / 3 = 2 : even
8C4 = 70 : even -> (8 – 4 + 1) / 4 = 5/4 : rational number

7C1 = 7 : odd -> (7 – 1 + 1) / 1 = 7 : odd
7C2 = 21 : odd -> (7 – 2 + 1) / 2 = 3 : odd
7C3 = 35 : odd -> (7 – 3 + 1) / 3 = 5/3 : rational number

If think of combinations as continuous of product,
it can see strange properties. Particularly, the part in bold.

コンビネーションを連続積で考えると、興味深い性質をみることができます。
特に、太字の部分です。

The continuous product of combinations seems to be useful for increasing k,
becoming even numbers, rational numbers, and so on.

コンビネーションの連続積は、kを上げていき、
偶数になる、有理数になるなどの判定に役立ちそうですね。