A shopkeeper keeps pens of two different companies

The number of ways of choosing two pens from 8 pens (assuming that there is one pen of each color) is 8!/(2!*6!)=28
In 4 cases, the color of the pens will be similar. So the number of ways of choosing 2 pens of different color is 28-4=24. The answer should be C)

A shopkeeper keeps pens of two different companies – Company A and Company B. Each company produces distinct pens and in 4 different colours: Red, Green, Blue, and Black and the shopkeeper keeps all these colours of both companies. James came to this shop to buy 2 pens, in how many ways can he buy pens of different colour?

A.6

B.12

C.24

D.40

E.48

Solution:

No of ways of selecting 2 pens of different colour out of 4 colours is 4C2 = 6 ways.

For each colour there are two companies ( A and B ) hence 2 options

So total number of ways will be 6 X 2 X 2 = 24

Hence, C is the correct answer.

We can choose 2 pens from A or 2 Pens from B or 1 each from A and B

Choosing any 2 pens from A = 4C2 = 6

Choosing any 2 pens from B = 4C2 = 6

1 From A and 1 from B = 4C1 * 3C1 = 12

Total possibilities = 6 + 6 + 12 = 24


Option C

Arun Kumar

Wrong. You need to divide this solution by 2. As we are just selecting 2 pens - order doesn't matter. In the 48 ways that you've found out, you'll find pairs like (R1,G2) and (G2,R1). Hence, the correct answer will be 48/2 = 24 (Option C)

The Pool of distinct items is small enough to Brute Force and find the different Combinations of 2 different colored pens.


The shop has 8 Distinct Pens:

4 colors from company A and the same 4 colors from company B


Co. A:
Red = Ra
Green = Ga
Blue = Ua
Black = Ka

Co. B:
Rb
Gb
Ub
Kb

You can list out each case of 2 different colors to see all the different scenarios (or find how many ways to choose 2 different colors out of 4 unique colors = “4 choose 2” = 6 ways)


Scenario 1:
Red: can get Ra or Rb ——2 available options

AND

Green: can get Ga or Gb ——- 2 available options

2 * 2 = 4 ways



OR



Scenario 2:
Red: can get Ra or Rb——-2 available options

AND

Blue: can get La or Lb ——-2 available options

2 * 2 = 4 ways



OR



Other 4 scenarios:

Red AND Black
Green AND Blue
Green AND Black
Blue AND Black


As you can see for each of the 6 scenarios in which he gets 2 different colored pens————> there are 4 Unique Pairings/Combinations/Ways to do so

(6 scenarios) * (4 Ways for each Scenario) =


24 total unique ways

-C-

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