Combinatorics

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Problem: The city council consists of a mayor and six elders. How many different four-member commissions can be formed if the mayor is in each commission? What if the mayor is not a member of any commission?

6.6.1 Combinatorics

1.52 $

Problem: In how many ways can six people choose from six gloves, right and left, so that none of them gets a pair?

6.6.2 Combinatorics

2.03 $

Problem: How many different ten-digit numbers can be written using these three digits \( 1,2,3 \), if digit 3 is used exactly twice in each number? How many of these numbers are divisible by 9 ?

6.6.3 Combinatorics

3.05 $

Problem: How many solutions in natural numbers does the system have: \[ \left\{\begin{array}{l} x+y=1999 \\ u+v=2000 \end{array}\right. \]

6.6.4 Combinatorics

1.52 $

Problem: How many diagonals are there in a convex 1999-gon?

6.6.5 Combinatorics

1.27 $

Problem: The platoon has 3 sergeants and 30 soldiers. How many ways are there to assign one sergeant and three soldiers to patrol?

6.6.7 Combinatorics

0.76 $

Problem: How many positive numbers between 20 to 1000 are divisible by exactly one of the numbers 7,11 or 13 ?

6.6.8 Combinatorics

1.52 $

Problem: Car license plates in this region consist of 3 digits (the total number of digits is 10) and three letters of the alphabet \( X=\{A, B, C, D, E, H, K, M, O, P, T, X, Y\} \). How many cars can be numbered with different numbers?

6.6.9 Combinatorics

1.27 $

Problem: There are \( n \) lines drawn on the plane, among which there is not a single pair of parallel lines and not a single triple of lines intersecting at one point. How many parts is the plane divided into by these lines?

6.6.10 Combinatorics

3.05 $

Problem: How many integers are there in the range 0 to 100000 , that contain exactly one digit 7 and one digit 3 ?

6.6.12 Combinatorics

3.05 $

Problem: Find the coefficients of \( a=x^{3} y^{2} z^{2}, b=x^{2} y^{2} z^{2} \), \( c=x^{4} z^{4} \) in the expansion \( \left(2 x+3 y+5 z^{2}\right)^{6} \).

6.6.13 Combinatorics

1.52 $

Problem: Prove that the binomial coefficient \( C(n, k) \) increases in \( n \) for fixed \( k \).

6.6.11 Combinatorics

1.27 $

Problem: Find the number of 7-digit telephone numbers in which the digit 4 appears and at least one of the first four digits is 9 .

6.6.14 Combinatorics

2.54 $

Problem: Prove the statement by mathematical induction: \( \left(n^{5}-n\right) \) is a multiple of 5 for all natural \( n \).

6.6.15 Combinatorics

0 $

Problem: Nine students must take a test in four subjects: Physics, Algebra, English and History. All tests are scheduled at the same time and each person can take only one test, so students need to be divided into groups. In how many ways can this be done? In how many ways can they take a sit after the test at two completely identical tables (at least one at a table) in order to celebrate the results?

6.6.16 Combinatorics

3.05 $

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