Rings 250 C 7. Show whe R oN2 (a +b12la, b E Q zt il la + bila, bE Z Q 8. Sho imp 9. Prov Z. is a 10. Ver 7Z 3Z 2Z 5Z 11. Pro 12. Let 9Z 6Z 4Z 10Z a is 13. De 18Z 12Z 8Z 14. Le Figure 12.1 Partial subring lattice diagram of C. m 15. Sh In the next several chapters, we will see that many of the fundamen- ri in group theory can be naturally extended to rings. In par- of tal concepts ticular, we will introduce ring homomorphisms and factor rings 16. S n 17. S Exercises 18. L There is no substitute for hard work. THOMAS ALVA EDISON, Life 19. 1. Give an example of a finite noncommutative ring. Give an example of an infinite noncommutative ring that does not have a unity 2. The ring (0, 2, 4, 6, 8) under addition and multiplication modulo 10 has a unity. Find it. 3. Give an example of a subset of a ring that is a subgroup under addition but nota subring. 20. 21. 22. 4. Show, by example, that for fixed nonzero elements a and b in a ring, the equation ax does this compare with groups? b can have more than one solution. How 23 5. Prove Theorem 12.2. 6. Find an integer n that shows that the rings Z, need not have the fol- lowing properties that the ring of integers has. a. a = a implies a = 0 or a = 1. b. ab 0 implies a = 0 or b = 0. c. ab ac and a 0 imply b c. Is the n you found prime? 25 26 27 2 224

Q: 3 -4- -0- 4. 2.

A:

Q: How many terms are there in the multinomial expansion of (x₁ + x₂ + ...........+xr)n?

A: Multinomial expansion :  (X1 + X2 + X3 + ......Xr)n  We have to calculate no. of terms in this…

Q: A = 6 -2 -2 10 -3 5 1 ---3-0 C=0-2 a = 1, b = [3 08]. -10 1 . B 5 2 11 9 4 4 7 0 0 11

A: We have to solve given Questions.

Q: 12 5(.8)0

A: We can solve given problem by using G.P series.

Q: 12 Find the term of (a – 3b)" that has b in it.

A:

Q: 2) 36³x=¹

A:

Q: FIND S + 4 (s² – 8S + 6) -

A:

Q: expand (a-b)7

A: The solution is given in the next step.

Q: Find the term indicated in the expansion (x + 2y)6; third term.

A: Find the term indicated in the expansion (x + 2y)6; third term.

Q: Finding the LCM of 4,5

A: Given,            LCM of 4,5.

Q: What is the coefficient of a b³c² in the expansion of

A: In this question, we need to find the coefficient of a4b3c2.

Q: 10. Find the sum of the coefficients of (2x– 3)°. c. -6560 d. -5120 а. -5660 b. -6780

A:

Q: 4. Fill in the following contingency table with the number of observations that fall into each com-…

A: x1 = number of colored oranges from Skittles brand = 77 x2 = number of colored oranges from M &…

Q: (n + 4)3 =

A: Given query is to expand (n+4)3

Q: 6/9 +  5/18 + 1 3/6

A:

Q: Expand the expression for Mr Taylor.

A: The previous solution is incorrect. Please follow this one.

Q: What is the 5th term in the expansion of (2x²y³ – ¹y)° O 14.25x7 y 20 15.75x10 y 19 15.05x¹0y17…

A: Given: 2x2y3-14y9=2x2y3-y49

Q: 18(9+7)+3

A: 18(9+7)+3

Q: 5. Find the coefficient of a¹¹b4 in the expansion of (a + b)¹5.

A:

Q: 29 Using the combination formula expand (5+x)

A:

Q: 4. Find the 11th term in the expansion of 25 3x + 5xi

A:

Q: Find the nth term of  (– 4 3 14 29 ) and (4, 5, 8, 13)

A: Sequence

Q: expand (x+y)6

A:

Q: 3 3/4 + 2 7/28

A:

Q: 3(m – 5) + m

A: 3(m-5)+m=0 3m-15+m=0 add both side 15 we get 2m=15

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.