Equation? The number of nonnegative integer solutions of is:
0002: From different elements? What can appear at most? ? The number of combinations of two elements is:
Throw? ? Dice, what is the sum of points? ? The number of combinations of is:
0003: From? ? From different elements? ? The number of combinations of nonadjacent elements is:
0004: Combination number formula:, from which we can deduce:
0005: Combination number formula:
0006: Combination number formula:
It can be inferred from this formula that:
0007: ? From coordinate system? ? The point moves along two axes to? ? At a point of the shortest path (where? ), the coordinates of passing points are always satisfied? ? The number of paths for is:
This conclusion can solve the following problems:
The ticket price of the cinema is 50 yuan. A man with 50 bucks? Individuals hold 100 yuan, each person only buys one ticket and does not borrow money from each other. As soon as the ticket office opened, there was no money. There are several queuing methods that can make the ticket sales smooth without change:
The probability of successful ticket sales is:
The minimum probability is:
0008: factorial formula:
0009: Combinatorial number function? , first increase and then decrease. Where is the function? ? Or? ? Get the maximum value.
00 10: number of iterations and combination:? Put a different ball in? ? In a box, in every box? ? A small ball, in which? , and then:
If the box is marked, the number of stacking schemes is:, and there are:
When each ball is put in? An option)
If this box is not marked, the number of stacking schemes is:
00 1 1: Proof:? The number of positive factors is odd;
Method 1:
According to fundamental theorem of arithmetic, there must be a prime number? ? And non-negative integers, so:
Can you see it? ? Can be broken down into. ? Product of prime factor power, so? ? The factor of is:
This is an odd number.
Method 2:
Settings? , is it? ? A positive number and a natural number? Are you satisfied? .
Can you see it? ? Appear in pairs, and there are even numbers of such positive factors.
? What else? Positive factors of, so the total number of all positive factors is odd.
Complete the certificate.
00 12: Proof:
?
Proof method using generating function:
Settings? , there is a recursive relationship:
? ;
Settings? , you can get the recurrence relation:
? ;
Settings? ? The parent function of is:
According to? The recurrence relation of can be obtained:
?
By solving the differential equation, we can get:
Therefore,
Complete the certificate.
00 13: proof: all positive integers? ? Can be expressed as the sum of (subscript) different Fibonacci numbers.
Prove by mathematical induction. When the proposition is clearly established, suppose? ? When the proposition is established, namely:
There are natural numbers? , satisfy:, then:
, the analysis is as follows:
What if? , the proposition is established; What if? And then what? , and:
What if? , the proposition is established; What if? And then what? , and:
What if? , the proposition is established; What if? And then what? , and:
?
And so on, this analysis will continue until:
, in which,, then:
What if? , the proposition is established; What if? , there are:
At this time, the proposition is established.
But from above, when? ? The same is true of the proposition of time.
Complete the certificate.
00 14: set? ? It's Fibonacci sequence. Is a positive integer, which proves that:
Solution: Use recursive relations? ? On the right side of the reducible equation is:
?
?
Will it be a general formula? ? Substitute the above formula and use the relation.
? ? Available:
?
?
?
?
Complete the certificate.
00 15: serial integer? ? Divide at will? ? And then what? ? In the two parts, it is proved that some of these two parts must contain three numbers that form an equal proportion relationship.
Solution: Reduce to absurdity. Hypothetically? ? And then what? ? Does not contain numbers that can form an equal proportion relationship. ? These three numbers can't belong to the same part, which will be discussed separately below.
1. If? And then what? , there are:
?
?
?
?
?
2. What if? And then what? , there are:
?
?
?
?
?
3. What if? And then what? This is discussed in two cases.
3. 1 What if? ,, have:
?
?
3.2 What if? ,, have:
?
?
?
To sum up, contradictions are introduced in all cases, so the hypothesis is incorrect and the proposition is proved.
00 16: It is proved that subgroups of cyclic groups are also cyclic groups.
Proof: setting? ? It is a cyclic group. what's up ? Generator? what's up ? Subgroups of.
Really? , set?
The following proof? ? All the elements are. ? Integer power of.
Hypothetically? ,
And then there's: that is? ? And then what?
What does it matter? ? Contradictory, so? ? what's up ? A generator, so? ? It is also a cyclic group, and the proposition is proved.
00 17: It is proved that the order of an element in a finite group is divisible by the order of the group.
Proof: setting? ? For finite groups, what's up ? Any element in the.
Yi Zheng? ? what's up ? A subgroup of, if the order of any subgroup is divisible by the order of the group, the problem is proved, and the facts are proved below.
Settings? ? what's up ? Any subgroup of. what's up ? Any two elements of.
What if? And then what?
Proof: Hypothesis? And then what? , and this? conflict
Proof: Hypothesis? And then what? , and this? ? conflict
Proof: If? , the proposition holds.
What if? And then what?
?
?
Therefore,
Settings? ,, then
Prove:
?
?
Therefore,
To sum up, available?
Settings? And then what?
Settings? And then what?
By analogy, you can get
Namely. ? What's the order? ? Integer multiples of, from which the following conclusions can be drawn:
The order of any subgroup of 1. finite group is divisible by the order of that group;
2. The order of any element of a finite group is divisible by the order of the group;
The proposition is proved.
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