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# Combinatorics | Number Theory Answers

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A cricket team of 11 players is to be selected from two groups of 6 and 8 players. In how many ways can the selection be made so that at least 4 players are taken from the group of 6.
A theater has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each succesive row of seats has two more seats in it than previous row

a) Calculate the number of seats in the 20th row.

b) calculate the total number of seats
Let U={-8,-7,....1,0,1,.....12}
A={x=-2<x<3}
B={x:-8<x<5}
C={x:0<x<12}
Find:
1.AUB
2.ANB
3.A
4.BNA
Use the fact that nCk equals n!/k!(n-k)! to express in factorials
1) The coefficient "u" of x^n in the expansion of (1+x)^2n
2) the coefficient "v" of x^n in the expansion of (1+x)^2n-1, Hence show that u=2v
Find the coefficient of x^6 in the expansion of (1-3x)(1+2x)^9 using the general term: t= nCr a^n-r b^r

1. Use mathematical induction to show that 8 │ (5^2n + 7).
Hint: 5^2(k+1) + 7 = 5^2(5^2k + 7) +(7 - 5^2·7)

2. Use the Division Algorithm to establish that 3a^2 – 1 is never a perfect square.

3. Use the Euclidean Algorithm to obtain integers x and y satisfying
gcd(1769,2378) = 1769x + 2378y.

4. Determine all solutions in the positive integers of the following Diophantine equation:
123x + 360y = 99.

5. Find the prime factorization of integer 1234, 10140, and 36000.

6. Give an example of a^2 ≡ b^2 (mod n) need not imply a ≡ b (mod n).

7. Show the following statements are true:
a. For any integer a, the unit digit of a^2 is 0, 1, 4, 5, 6 or 9.
b. Any one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occur as the units digit of a^3.
c. For any inter a, the units digit of a^4 is 0, 1, 5 or 6.

The number of letters in the language of a weird country is 5 and no one in that
country uses more than 3 letters to make a word. What is the highest number of
words one can make in that language?
An optimization model includes a chance constraint to satisfy demand of a particular product. The demand is uncertain and is modeled with an integer uniform distribution with parameter value of 0 and 4. That is, the probability that the demand is 0, 1, 2, 3, or 4 is exactly the same. A decision is made to order 2 units of the product from a supplier in order to satisfy the uncertain demand. What is the value at risk (VaR) for the demand constraint?
Prove that $\sum_{i=1}^s a_i=2^{n}-1-\sum_{i=s+1}^n a_i=2^s-1. Note that$\sum_{i=1}^n=2^n-1.
INTEGERS AND DIVISIBILITY CONCEPTS

1.Show that if d ≠ 0, then d | (-a) and -d | a.
2.Show that it is false that a>b implies a|b.
3.Is 980637 divisible by 7? Show.
4.Determine whether of the following are divisible by 3, 5, 7, 9, or 11 using the methods described int he text:
A. 1969
B. 28350
C. 1421
D. 17303
E. 116424
F. 1089
5.Classify each of the following as true or false:
A. 6 is a divisor of 24.
B. 40 is a multiple of 8.
C. 0 divides 10.
D. 13 is a factor of 33.
E. 12 divides 6.
6.Show that 23n -1 is divisible by 7.
7.Show that 5n - 1 is divisible by 4.

GREATEST COMMON FACTOR (GCF) AND THE LEAST COMMON MULTIPLE (LCM)

1.Calculate (3141 , 1592).
2.Find x and y such that, 3141x + 1592y =1.
3.Find the solution of 803x + 154y = 33.
4.Find the GCD and LCM of the numbers 63,24, 99.

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