Answer with explanation:
→Number of Integers from 1 to 100
=100(50 Odd +50 Even)
→50 Even =2,4,6,8,10,12,14,16,...............................100
→50 Odd=1,3,,5,7,9,..................................99.
→Sum of Two even integers is even.
→Sum of two odd Integers is odd.
→Sum of an Odd and even Integer is Odd.
(a)
Number of ways of Selecting 2 integers from 50 Integers ,so that their sum is even,
=Selecting 2 Even integers from 50 Even Integers , and Selecting 2 Odd integers from 50 Odd integers ,as Order of arrangement is not Important, ,
![=_{2}^{50}\textrm{C}+_{2}^{50}\textrm{C}\\\\=\frac{50!}{(50-2)!(2!)}+\frac{50!}{(50-2)!(2!)}\\\\=\frac{50!}{48!\times 2!}+\frac{50!}{48!\times 2!}\\\\=\frac{50 \times 49}{2}+\frac{50 \times 49}{2}\\\\=1225+1225\\\\=2450](https://tex.z-dn.net/?f=%3D_%7B2%7D%5E%7B50%7D%5Ctextrm%7BC%7D%2B_%7B2%7D%5E%7B50%7D%5Ctextrm%7BC%7D%5C%5C%5C%5C%3D%5Cfrac%7B50%21%7D%7B%2850-2%29%21%282%21%29%7D%2B%5Cfrac%7B50%21%7D%7B%2850-2%29%21%282%21%29%7D%5C%5C%5C%5C%3D%5Cfrac%7B50%21%7D%7B48%21%5Ctimes%202%21%7D%2B%5Cfrac%7B50%21%7D%7B48%21%5Ctimes%202%21%7D%5C%5C%5C%5C%3D%5Cfrac%7B50%20%5Ctimes%2049%7D%7B2%7D%2B%5Cfrac%7B50%20%5Ctimes%2049%7D%7B2%7D%5C%5C%5C%5C%3D1225%2B1225%5C%5C%5C%5C%3D2450)
=4900 ways
(b)
Number of ways of Selecting 2 integers from 100 Integers ,so that their sum is Odd,
=Selecting 1 even integer from 50 Integers, and 1 Odd integer from 50 Odd integers, as Order of arrangement is not Important,
![=_{1}^{50}\textrm{C}\times _{1}^{50}\textrm{C}\\\\=\frac{50!}{(50-1)!(1!)} \times \frac{50!}{(50-1)!(1!)}\\\\=\frac{50!}{49!\times 1!}\times \frac{50!}{49!\times 1!}\\\\=50\times 50\\\\=2500](https://tex.z-dn.net/?f=%3D_%7B1%7D%5E%7B50%7D%5Ctextrm%7BC%7D%5Ctimes%20_%7B1%7D%5E%7B50%7D%5Ctextrm%7BC%7D%5C%5C%5C%5C%3D%5Cfrac%7B50%21%7D%7B%2850-1%29%21%281%21%29%7D%20%5Ctimes%20%5Cfrac%7B50%21%7D%7B%2850-1%29%21%281%21%29%7D%5C%5C%5C%5C%3D%5Cfrac%7B50%21%7D%7B49%21%5Ctimes%201%21%7D%5Ctimes%20%5Cfrac%7B50%21%7D%7B49%21%5Ctimes%201%21%7D%5C%5C%5C%5C%3D50%5Ctimes%2050%5C%5C%5C%5C%3D2500)
=2500 ways