Respuesta :
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, ,
[tex]=_{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[/tex]
=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,
[tex]=_{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[/tex]
=2500 ways
The number of ways for the sum to be even is 2450 and the number of ways for the sum to be odd is 2500.
What is permutation?
A permutation is an act of arranging the objects or elements in order.
The number of integers from 1 to 100 is 100.
50 Even = 2,4,6,8,10,.........100
50 odd = 1,3,5,7,9,.........99
We know that the sum of two even numbers is even and the sum of two odd numbers is also even but the sum of even and odd numbers is odd.
a. Number of ways of selecting 2 integers from 50 integers, so their sum is even.
Selecting 2 even integers from 50 even integers and selecting 2 odd integers from 50 odd integers, as the order of arrangement is not important. then
[tex]\rm ^{50}C_{2} + ^{50}C_{2}\\\\\dfrac{50!}{48!*2!} + \dfrac{50!}{48!*2!}\\\\\\\dfrac{50*49}{2} +\dfrac{50*49}{2}\\\\2450[/tex]
The number of ways is 2450.
b. Number of ways of selecting 2 integers from 100 integers, so their sum is odd.
Selecting one from 50 even numbers and selecting one from 50 odd numbers. Then
[tex]\rm ^{50}C_1 * ^{50}C_1\\\\\dfrac{50!}{49!*1!} * \dfrac{50!}{49!*1!}\\\\50 * 50\\\\2500[/tex]
The number of ways is 2500.
More about the Permutation link is given below.
https://brainly.com/question/1216161
