Q1:
1st digit: 4 possibilities
2nd digit: 5 possibilities (dont forget 0)
4th digit: 5 odd numbers, so 5 possibilities
3rd and 5th: 4 possibilities each

64 * 25 = 1600

Q2:
2/5. Could someone post explanation of the numeric solution, because I solved it my drawing table and fixing one of the choices?

1st Q:
One 4
1*4*5*5*5 = 500
3*1*5*5*5 = 375
No 4
3*4*5*5*5 = 1500
sum = 2375

2nd Q:
(2*4!)/5! = 2/5

Answer 1:
E-Even
O- ODD
Number is of the form EE OOO
no 4's
3*4*5*5*5=1500
One 4:
when 4 at first place:
4*5*5*5=500
when 4 at 2nd place:
3*5*5*5=375
total=2375
Answer C

Answer 2)
circular permutation:
total number ways of arranging 6 people in circular arrangement=5!
when two people are adjacent=4!*2
probability=4!*2/5!= 2/5
Answer B

Q 1. - E.
Expected format: EEOOO
With 4 as a possibility for first E (Then second E cannot be 4):-

4 * 4 * 5 * 5 * 5 = 16 * 125.

With 4 as a possibility for second E(Then first E cannot be 4):-

3 * 5 * 5 * 5 * 5 = 15 * 125.

Total = 31 * 125 = 1875.

Q 2. - B.
Number of ways of selecting 2 people out of 6:
6C2 = 15.

Imagine 6 people sitting on a round table. Then you will have 6 pairs of adjacent people sitting (Instead of 5 since this is a round table), Hence, total number of adjacent people "at a time" = 6

Probablity = 6 / 15 = 2/5.