# Personalized License Plate: Part 1

MTH109 – Mathematical Explorations

Colorado State University – Global Campus

Assuming repetitions are allowed, how many custom tag numbers are possible?

C(n, r) = (r + n – 1)! / (r! × (n – 1)!)

n = 36 and r = 6

Find (r + n – 1)!

(r + n – 1)! = (6 + 36 – 1)! = 41!

41!= 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30×31×32×33×34×35×36×37×38×39×40×41

= 3.3452526613163803e+49

Find r!

6! = 1×2×3×4×5×6 = 720

Find (n – 1)!

(n – 1)! = (36 – 1)! = 35!

35! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30×31×32×33×34×35

= 1.0333147966386144e+40

C(n, r) = (r + n – 1)! / (r! × (n – 1)!)

C(36, 6) = 3.3452526613163803e+49 / (720 × 1.0333147966386144e+40)

C(36, 6) = 4,496,388 = number of custom tag numbers with repetitions

Assuming repetitions are not allowed, how many custom tag numbers are possible?

C(n, r) = n! / (r! × (n – r)!)

n = 36 and r = 6

Find n!

36! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30×31×32×33×34×35×36

= 3.719933267899012e+41

Find r!

6! = 1×2×3×4×5×6 = 720

Find (n – r)!

(n – r)! = (36 – 6)! = 30!

30! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30

= 2.6525285981219103e+32

C(n, r) = n! / (r! × (n – r)!)

C(36, 6) = 3.719933267899012e+41 / (720 × 2.6525285981219103e+32)

C(36, 6) = 1,947,792 = number of custom tags with no repetitions

Assuming repetitions are not allowed and there must be exactly three letters, how many custom tag numbers are possible?

n = 6 and r = 3

C(n, r) = n! / (r! × (n – r)!)

Find n!

6! = 1×2×3×4×5×6 = 720

Find r!

3! = 1×2×3 = 6

Find (n – r)!

(n – r)! = (6 – 3)! = 3!

3! = 1×2×3 = 6

C(n, r) = n! / (r! × (n – r)!)

C(6, 3) = 720 / (6 × 6)

C(6, 3) = 20 = number of combinations where 3 of the positions will be letters

Letters that can be in any position, so long as there are 3 =

263 = 17,576

Numbers that can be in any position, so long as there are 3 =

103 = 1,000

20×17,576×1,000 = 351,520,000 = custom tags with exactly 3 letters

Assuming repetitions are not allowed and there must be exactly three letters, located at the beginning of the license plate, how many custom tag numbers are possible?

P(Ln,r)×P(Nn,r)

P(26,3)×P(10,3)

Letters

P(n, r) = n! / (n – r)!

Find n!

26! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26

= 4.0329146112660565e+26

Find (n – r)!

(n – r)! = (26 – 3)! = 23!

23! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23 = 2.585201673888498e+22

P(n, r) = n! / (n – r)!

P(26, 3) = 4.0329146112660565e+26/2.585201673888498e+22

P(26, 3) = 15600

Numbers

P(n, r) = n! / (n – r)!

Find n!

10! = 1×2×3×4×5×6×7×8×9×10 = 3628800

Find (n – r)!

(n – r)! = (10 – 3)! = 7!

7! = 1×2×3×4×5×6×7 = 5040

P(n, r) = n! / (n – r)!

P(10, 3) = 3628800/5040

P(10, 3) = 720

P(26,3)×P(10,3) = 15600×720 = 11,232,000 = custom tags with 3 letters at the beginning

Assuming repetitions are not allowed, there must be exactly three letters, located at the beginning of the license plate, and that the letters can only be vowels, how many passwords are possible?

P(Vn,r)×P(Nn,r)

P(5,3)×P(10,3)

Vowels

P(n, r) = n! / (n – r)!

Find n!

5! = 1×2×3×4×5 = 120

Find (n – r)!

(n – r)! = (5 – 3)! = 2!

2! = 1×2 = 2

P(n, r) = n! / (n – r)!

P(5, 3) = 120/2

P(5, 3) = 60

Numbers

P(n, r) = n! / (n – r)!

Find n!

10! = 1×2×3×4×5×6×7×8×9×10 = 3628800

Find (n – r)!

(n – r)! = (10 – 3)! = 7!

7! = 1×2×3×4×5×6×7 = 5040

P(n, r) = n! / (n – r)!

P(10, 3) = 3628800/5040

P(10, 3) = 720

P(5,3)×P(10,3) = 60 × 720 = 43,200 = custom tags with exactly 3 vowels