Please only answer (g) and (h) (in image), I'm providing the answers to the other questions below. (3) In the part (a) of this question, I’m asking you to give a background situation for the combinatorics (“counting”) problems in the rest of the question. In parts (b) through (h), you’ll give a question based on that situation for which the calculation I’ve given you
Please only answer (g) and (h) (in image), I'm providing the answers to the other questions below.
(3) In the part (a) of this question, I’m asking you to give a background situation
for the combinatorics (“counting”) problems in the rest of the question. In parts (b) through
(h), you’ll give a question based on that situation for which the calculation I’ve given you
would be the correct answer.
a)
Combinations and Permutation
When one has to choose among a set of objects or people, then combination is used to find the number of ways. The order in which they are chosen does not matter here.
On the other hand, if one has to find the number of ways of arranging certain objects, then arrangement is used. Here, the order always matters.
For example, find the number of ways in which 4 benches can be occupied by 4 students. Here, the first bench is occupied by 4 students, second one with the remaining 3 students, third one with remaining 2 students, and 4th bench with the last student. Thus, this is permutation as the order matters here.
Now, suppose there are 5 seats and the number of ways of choosing 4 seats for 4 students can be chosen by using combinations. Here, the order does not matter because it does not matter which student is occupying which seat.
Formula for permutation is:
- When repetition is allowed: nCr
- When repetition is not allowed: nPr = n! / (n−r)!
Formula for combination is: nCr = n! / r! * (n−r)!
(b) 19! = 362,880
(c) P(19, 10) = 335,221,286,400
(d) 196 = 47,045,881
(e) C(19, 12) = 50,388
(f) C(8, 5) × C(11, 7) = 18,480
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