二項定理

1 ：１３２人目の素数さん：2017/04/30(日) 12:23:52.89 ID:0Ljtscid.net

証明の方法を誰か教えてくれないか

2 ：１３２人目の素数さん：2017/04/30(日) 13:32:45.55 ID:gRkPmJOD.net

二項展開の公式なら、よく知っているけれど、
改めて聞かれてみると、定理としての「二項定理」って
公式のどの部分を呼んでいるんだろうね。
(x,y)^n が x,y の n 次同次式であることや、
任意の多項式が積和形に整理できることは自明として、
二項展開の係数として定義する二項係数 (n k) と
n 個の中から k 個取り出す組み合わせ nＣk と
n!/(k!(n-k)!) のどれとどれが等しいことを
二項定理と呼ぶのか？

証明を書くとなると、nＣk を既知として
nＣk が満たす漸化式から (n k) = nＣk を
示している本が多いように思うが、
本によっては、そもそも nＣk を二項係数の記号として導入
しているから、それではマズいような気もする。

立場１
nＣk は、n 個の中から k 個取り出す組み合わせ
として定義する。
(x,y)^n　= (x+y)(x+y)…(x+y) の右辺を展開して
(x^k)y^(n-k) がどこに現れるかを見れば、
(n k) = nＣk が判る。
この立場では、nＣk の定義が散文的なので、
証明も説明的、非形式的にならざるを得ない。

立場２
nＣk を = n!/(k!(n-k)!) として定義する。すると
漸化式 nＣk = (n-1)Ｃ(k-1) + (n-1)Ｃk が示せるので、
これを使って、n についての数学的帰納法で
(x+y)^n = Σ[k=0…n](nＣk)(x^k)y^(n-k) を示す。

立場３
nＣk を全く用いずに、
(x+y)^n = Σ[k=0…n]{n!/(k!(n-k)!)(x^k)y^(n-k) を
n についての数学的帰納法で示す。
やっていることは立場２と同じで、nＣk を表面上伏せるために
漸化式を示す部分を帰納法の中に直接展開してしまっただけ。
この立場だと、nＣk を = (n k) として定義しても問題ない。

中間的なものとして、立場１の下で nＣk = n!/{k!(n-k)!}
を示して立場２の証明に合流する方法もある。

3 ：￥ ：2017/05/08(月) 19:49:31.26 ID:OR+quqWp.net

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4 ：￥ ：2017/05/08(月) 19:49:55.50 ID:OR+quqWp.net

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9 ：￥ ：2017/05/08(月) 19:51:49.90 ID:OR+quqWp.net

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10 ：￥ ：2017/05/08(月) 19:52:06.69 ID:OR+quqWp.net

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11 ：￥ ：2017/05/08(月) 19:52:30.85 ID:OR+quqWp.net

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12 ：￥ ：2017/05/08(月) 19:52:53.40 ID:OR+quqWp.net

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13 ：１３２人目の素数さん：2017/05/09(火) 22:03:06.86 ID:Yy9PTPKs.net

ゴミスレで悲惨な荒らし

14 ：１３２人目の素数さん：2017/05/25(木) 11:17:33.54 ID:ru5RQGVh.net

It is a pity that some men look down on
women.
It is rude for a man to ask a woman how old
she is.
I will buy this.
It is well-known that smoking is bad for your
health.
You should know better than to disturb others
in the library.
have a good command of.
your conduct, your actions.
bother others, annoy others, disturb others,
cause trouble to other people.
Tom went to Germany two years ago.
I lived in China for three years when I was
a child, but I cannot speak Chinese at all.
I have decided to go to Germany next year.
Tom has been to Germany three times.
just now, ago, ever.
lend me the book, lend the book to me,
borrow the book from me,
lend lent lent, rent rented rented
Neither of my parents is interested
in computers.
i do not know either of the two girls.
i do not know both of the two girls.
None of the three girls is interested
in computers.
i do not know any of the three girls.
i do not know all of the three girls.
several pieces of science-fiction.
The teacher told me to hand in this paper
by Tuesday.
I have still not read, I have not yet read.
a physical examination.

15 ：１３２人目の素数さん：2017/05/25(木) 11:28:22.19 ID:ru5RQGVh.net

One of the purposes of a college education is
to create people who can think on their own.
One of my pleasures on holidays is
fishing in the nearby river.
The trouble with you is that you do not listen
to other people.
To win a girl's heart, all David has to do
is smile at her.
the fact is that, the point is that,
the trouble is that, All you have to do is
write with a pencil, write in pencil,
write by hand, write with my left hand
write on a word processor,
write on the word processor.
look at the table with the naked eye
walk with bare feet, walk barefoot.
I do not like to write letters by using
the computer.
You should be a little more considerate
of young people's feelings.
I cannot just move abroad.
i have to consider my family.
think about, think of.
tell jokes, Tom never tells lies.

I suddenly realized that I was walking
in the wrong direction.
Tom told me that he had lived in China
for two years.
It never occurred to me that
my remark might please her.
I crossed my fingers during the race
so that my son would win.
I asked Tom if anybody had called me
while I was out.
I want to know what the new teacher
is like.
I do not know what it is like
to live in the country.
I do not know how Tom is.
almost everything, almost anything,
almost nothing, almost all.

16 ：１３２人目の素数さん：2017/05/25(木) 13:10:32.02 ID:ru5RQGVh.net

I have lived in the suburbs of kyoto for
ten years. I have been riding this French
bicycle for more than ten years.
Prices have gone up by fifty percent in the
last ten years. Prices have gone up by fifty
percent over the last ten years. There have
been five major earthquakes in the last ten
years. / live, stay, know / learn, rain, sleep,
stand, study, teach, wait, work.
cultural heritage.
My father is a lawyer by profession.
Yesterday I bought the book I had long
wanted to get. Yesterday I could not buy
the book Tom had recommended. When
I was young, I could run like a cheetah!
I told him that I could swim ... , could hear,
could see, could feel, could understand,
my supervisor, my manager, my boss.
It seems that Tom takes very good care of
his car. Tom seems to take very good care
of his car. let me, allow me to.
I do not know how I am going to get
Susan to give up smoking, but I am going
to do it somehow. Somehow they
managed to climb in through the
bathroom window.

17 ：１３２人目の素数さん：2017/05/25(木) 13:18:32.25 ID:ru5RQGVh.net

Yesterday Tom went to the bank to withdraw
some money. I worked hard so that everything
would be ready in time. We just play poker for
fun, not for money. The book is used for
studying. Tom tried to get another job, but he
had no luck. I tried making a Spanish dish for
the first time. It was delicious. you should make
an effort to be polite to others.
my cherished memories, my cherished dreams,
my cherished possessions.
You should eat lunch so that you will not get
hungry on the way. You should take
sandwiches with you in case you get hungry
on the way. You should take care not to swear
in a job interview. I ran to the bus stop so that
I would not miss the bus. I ran to the bus stop
to be in time for the bus.
write, write wellwrite an essay.
In general, short sentences are easier to
understand than long ones.
a friend, friends, my friends, my friend, tom,
my friend. overlook, look over, check

18 ：１３２人目の素数さん：2017/05/25(木) 13:24:21.55 ID:ru5RQGVh.net

Japanese tourists find the western custom of tipping troublesome, because they do not know how much money they should leave. I rarely go out, mainly because I have to look after my child.
partly because I have to look after my child. You do not have to be discouraged just because you are not a genious. / probably because you are not a genious.
The accident was very tragic and I am afraid I just cannot bring myself to talk about it at the moment.
I am reluctant to talk about it, I do not feel like talking about it, I feel like talking about it.
Thanks to Mr. Suzuki, the party was so enjoyable that we lost track of time. It was such a cold day that I decided to stay indoors.
In Japan bowing is so familiar a thing in daily life that it is easy to take it for granted. She always buys such expensive clothes that I am not surprised that she never has any money.
such cold weather, so many cars, so much water, prices, house prices, land prices, oil prices, the price of books, a charge, a fare, a school fee, commodity prices
(expensive, inexpensive, cheap)
(high, low, reasonable) (large, small)
Property is so expensive...
Nancy spends a lot of money on famous brand name goods. young people today, students today

19 ：１３２人目の素数さん：2017/05/25(木) 13:47:37.45 ID:ru5RQGVh.net

The artist's signature proves that this painting
is genuine. Only three people passed the test,
which shows how hard it was. The test was
very difficult. This is proved by the fact that
only three people passed it. There is
evidence to show that the test was very
difficult. This is clear from the fact that only
three people passed it.
Our school is thirty years old. Ann looks
young for her age. I am eighteen years old.
I am eighteen. I am eighteen of age.
children of all ages, reach thirty, reach my
forties, build a male-oriented society,
become a member of society, live in the
global community, japan is the oldest country.
Life has certainly become easier, but this
does not necessarily mean that everyone is
happier than before. It does not necessarily
follow that everyone is happier than before.
If you forget what you have experienced,
this means that you have not experienced it
at all. it follows that you have not experienced
it at all. just because you are rich, it does not
necessarily follow that you are happy.
i know a lot about cars. know Tom very well,
know English, know the answer, learn the fact.
learn a lot about, learn little about, learn
something about, learn nothing about.
know a lot about, know little about, know
something about, know nothing about.
know that, know how, learn that, learn how.
This is the case in other countries

20 ：１３２人目の素数さん：2017/05/25(木) 14:08:58.50 ID:ru5RQGVh.net

The acidity of water is essentially determined
by the acidity of the soil. Your success in life
depends on your education. Children are
greatly influenced by the images parents
have of their future. depend too much on
instant foods, It is not good for the health to
eat too much instant food.
Tom has good taste in music. Do you have
any hobbies? What is your hobby? My
hobby is listenig to music. I like listenig to
music. Listenig to music is my favorite
pastime.
intellectual level, a high standard of living
When you have caught a cold, it is best
to go to bed at once. The best way to
enjoy natural beauty is to walk in a
leisurely way. The best thing about
teaching english is discoverng something
new about the japanese language.
When I was in college, met the woman
of my dreams.
I should have studied, I should not have
studied, I wish I had studied. I wish I had
not studied. I regret that I did not learn.
I regret that I learned.

21 ：１３２人目の素数さん：2018/06/24(日) 09:38:46.02 ID:CiX9DLBe.net

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