Combinatorics

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

6.6.17 Combinatorics

2.03 $

Problem: Find the sequence \( \left\{a_{n}\right\} \), that satisfies the recurrence relation \( 2 \cdot a_{n+2}+10 \cdot a_{n+1}+8 \cdot a_{n}=0 \) and the initial conditions \( a_{1}=3, a_{2}=9 \).

6.6.18 Combinatorics

0 $

Problem: How many positive three-digit numbers are there: a) divisible by the numbers 8,22 or 26 ? b) divisible by exactly one of these three numbers?

6.6.19 Combinatorics

3.05 $

Problem: Prove the statement using mathematical induction: (8 \( { }^{n}-1 \) ) is a multiple of 7 for all natural \( n \geq 1 \).

6.6.20 Combinatorics

0 $

Problem: A company of 8 people went fishing. In order to organize dinner and overnight stay, they need to prepare firewood, make a fire, prepare food, and put up tents. To complete all these tasks, it is necessary for them to split into groups "firemakers", “cooks", "house builders". a) How many different ways are there for such a division if no group should include less than 2 people? b) How many different ways are there to sleep in three completely identical tents, without being arranged one by one?

6.6.21 Combinatorics

3.05 $

Problem: How many positive three-digit numbers are there: a) not divisible by any of the numbers \( 5,15,25 \) ? b) divisible by exactly one of these three numbers?

6.6.22 Combinatorics

3.05 $

Problem: Find the coefficients when \[ a=x^{2} \cdot y^{4} \cdot z^{6}, b=x^{2} \cdot y^{2} \cdot z^{2}, c=x^{2} \cdot y^{8} \] in the expansion \( \left(4 \cdot x+5 \cdot y^{2}+z^{3}\right)^{6} \).

6.6.23 Combinatorics

2.03 $

Problem: Find a sequence \( \left\{a_{n}\right\} \), satisfying the recurrence relation \( 2 a_{n+2}-10 a_{n+1}+8 a_{n}=0 \) and the initial conditions \( a_{1}=0, a_{2}=-12 \).

6.6.24 Combinatorics

1.02 $

Problem: Prove the statement using mathematical induction: \[ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=n \cdot(n+1)(2 n+1) / 6 \text {. } \]

6.6.25 Combinatorics

1.02 $

Problem: Seven employees of the company are sent to study a foreign language, and they need to be distributed to study English, German and French (each learns only one language). a) How many different ways of such distribution are there? b) In how many ways can they be arranged to study in two completely identical library rooms (at least one in a room)?

6.6.26 Combinatorics

3.81 $

Problem: How many positive three-digit numbers are there: a) divisible by the numbers 5,18 or 21 ? b) divisible by exactly one of these three numbers?

6.6.27 Combinatorics

3.05 $

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

6.6.28 Combinatorics

2.03 $

Problem: Find a sequence \( \left\{a_{n}\right\} \), satisfying the recurrence relation \( 2 a_{n+2}-10 a_{n+1}+12 a_{n}=0 \) and the initial conditions \( a_{1}=3, a_{2}=27 \).

6.6.29 Combinatorics

1.02 $

Problem: Nine employees of the company are sent to study a foreign language. They need to be distributed to study English, Spanish, German and French (each learns only one language). a) How many different ways of such a distribution exist? b) In how many ways can they be arranged to study in three completely identical rooms of the library (at least one in a room)?

6.6.30 Combinatorics

3.05 $

Problem: How many positive three-digit numbers are there: a) not divisible by any of the numbers \( 9,14,21 \) ? b) divisible by exactly one of these three numbers?

6.6.31 Combinatorics

3.05 $

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