Question

find how many different numbers can be made by rearranging all 5 digits of the number 12345 if a) there is no restrictions. b) the number made is an even number. c) sandra wishes to buy some applications for her smart phone but she only has enough money fot 5 apps in total. there are 3 train apps, 6 social network apps and 14 game apps available. sandra wants to have at least 1 of each type of app. find the number of different possible selections of 5 apps that sandra can choose.

Answer 1

(a) When there is no restrictions

No. of different numbers will be = !5

As 1st place can be filled with any of the 5 digits

2nd place with any of the remaining 4 digits

and so on.

Thus total different numbers = 5*4*3*2*1 = 120

(b) If the number made need to be an even number, the unit place, that is the 5th place needs to be filled with an even number, 2 or 4 in this case.

Thus, 5th place can be filled in 2 ways

4th place can be filled with any of the remaining 4 digits in 4 ways,

3rd place with 3 digits in 3 ways,

2nd place with 2 digits in 2 ways

and 1st place with 1 digit in 1 way

Thus, total different even numbers = 2*4*3*2*1 = 48

(c)There can be two arrangements for selection of 5 apps out of three categories having atleast 1 of each type

(i) 1,1,3 with three different arrangements again

(ii) 1,2,2 with three different arrangements again.

Thus, total 6 cases

Case 1: 1 train, 1 social network, 3 game apps in ways = 6552

Case 2: 1 train, 3 social network, 1 game app in ways = 840

Case 3: 3 train, 1 social network, 1 game app in ways = 84

Case 4: 1 train, 2 social network, 2 game apps in ways = 4095

Case 5: 2 train, 1 social network, 2 game apps in ways = 1638

Case 6: 2 train,2 social network, 1 game apps in ways = 630

Thus, different number of possible selection of 5 apps = 6552 + 840 + 84 + 4095 + 1638 + 630 = 13839 ways