For this group of questions, assume repetitions are not permitted. (a) how many 3-digit numbers can be formed from the six digits 1, 3, 5, 7, 9, 0? (b) how many of these numbers are less that 500? (c) how many are even? (d) how many are odd? (e) how many are multiples of 5?

a. 100

b. 40

c. 20

d. 100

e. 40

Step-by-step explanation:

The given numbers are 1, 3, 5, 7, 9, 0

If the repetitions are not allowed,

(a) Let the three digit number is xyz.

At x number of digits that can be placed = 5 [except 0]

At y we can place digits = 5

At z number of digits we can place = 4

Total number of digits that can be formed = 5×5×4 = 100

(b). Numbers formed which are less than 500.

Let the number is in the form of xyz

Then x will be less than 5 to form numbers less than 500.

So possible number of digits at x = 2 [1, 3]

Possible number of digits at y = 5

Possible number of digits at z = 4

Total numbers formed = 2×5×4 = 40

(c) Even numbers will have 0 at the place of z.

Then number of digits at x may come = 5

Number of digits at y = 4

At z only one digit may come.

Therefore, number of even numbers formed = 5×4×1

= 20 numbers

(d) Odd number means at z there should be any one out of 1, 3, 5, 7, 9.

Therefore, at z number of digits may come = 5

At x number of digits may come = 5

At y number of digits may come = 4

Total odd numbers = 5×5×4 = 100

(e) Multiple of 5 means at z there may be either 0 or 5.

So numbers having 0 or 5 at z = 2

Now digits that may come at x = 5

Digits that may come at y = 4

Total numbers may be formed = 2×5×4 = 40

1. a^4+8a^3+24a^2+32a+16

2.y^8+8y^7+28y^6+56y^5+70y^4+56y^3+28y^2+8y+1

3.x^6-3x^4y^2+3x^2y^4-y^6