Ron wants to make sandwiches for him and his friends He has

Ron wants to make sandwiches for him and his friends. He has the following choices in breads (whole wheat, multigrain, rye). He has these types of meat choices {turkey, ham}. He has these types of choices for vegetables (lettuce, tomatoes). Finally, he has these choices for spreads (mayo, mustard). If each sandwich has bread, spread, meat, and vegetable, How many possible combination one are there? 4. Simplify using the Binomial Theorem (2a - 3b)^3.

Solution

3. Ron can choose 1 bread from 3 diffirent kinds in ³C₁=3 ways.

For each selection of bread Ron can choose one type of meat from two different types in ²C₁=2 ways.

So bread and meat can be chosen in 3x2=6 different ways.

Now for each selection of bread and meat he can choose one kind of vegetable from two different kinds in ²C₁=2 different ways.

Hence bread, meat and vegetable can be chosen in 6x2=12 different ways.

Again for each choice of bread, meat and vegetable Ron can choose one kind of mayo from two different kinds in ²C₁=2 ways.

Therefore bread, meat, vegetable and mayo can be chosen in 12x2=24 ways, i.e., there are 24 different possible combinations.

4. (2a-3b)³=(2a)³+³C₁(2a)²(-3b)+³C₂(2a)(-3b)²+(-3b)³

                 =8a³-36a²b+54ab²-27b³