Combinatorial selection with category constraints

Questions asking for the number of ways to select a specified number of items from multiple distinct categories where each category has a minimum or maximum constraint (e.g., select 2 pop CDs, 2 jazz CDs, and 1 classical CD).

3 questions · Moderate -0.9

5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems
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CAIE S1 2008 June Q3
6 marks Moderate -0.8
3 Issam has 11 different CDs, of which 6 are pop music, 3 are jazz and 2 are classical.
  1. How many different arrangements of all 11 CDs on a shelf are there if the jazz CDs are all next to each other?
  2. Issam makes a selection of 2 pop music CDs, 2 jazz CDs and 1 classical CD. How many different possible selections can be made?
OCR S1 2010 June Q7
8 marks Moderate -0.8
The menu below shows all the dishes available at a certain restaurant.
Rice dishesMain dishesVegetable dishes
Boiled riceChickenMushrooms
Fried riceBeefCauliflower
Pilau riceLambSpinach
Keema riceMixed grillLentils
PrawnPotatoes
Vegetarian
A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes and 4 different vegetable dishes from this menu. Given these restrictions,
  1. find the number of possible combinations of dishes that they can choose to share, [3]
  2. assuming that all choices are equally likely, find the probability that they choose boiled rice. [2]
The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.
  1. Find the number of possible combinations of dishes that they can now choose. [3]
OCR MEI S1 2010 January Q6
4 marks Easy -1.2
Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if
  1. no student may win more than 1 prize, [2]
  2. no student may win all 3 prizes. [2]