First slide

Combinations

Question

Given 5 different green dyes, 4 different blue dyes and 3 different red dyes, the number of combinations of dyes that can be chosen by taking at least one green and one blue dye is

Moderate

A

248
B

120
C

3720
D

465

Solution

Number of ways of selecting at least one green dye out of 5 different green dyes
$^{5} C_{1} +^{5} C_{2} +^{5} C_{3} +^{5} C_{4} +^{5} C_{5} = 2^{5} - 1$
Also, at least one blue dye can be selected out of 4 blue dyes in
$^{4} C_{1} +^{4} C_{2} +^{4} C_{3} +^{4} C_{4} = 2^{4} - 1$
Again, 3 different red dyes can be selected in
$^{3} C_{0} +^{3} C_{1} +^{3} C_{2} +^{3} C_{3} = 2^{3}$ ways
$∴ Required ways = (2^{5} - 1) (2^{4} - 1) (2^{3}) = 3720$

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