As the leader of a popular indie math rock group, you would like to hold a concert. A concert consists of a number of songs played in sequence, followed by a period of reflection. The reflection lasts 0 or more minutes, in which the musicians sit on stage staring at the audience, who are invited to meditate on the proceedings. You have the following songs available: “Faerie’s Aire and Death Waltz” (4m) “Duetto Boffo di Due Gatti” (4m) “Three Minute Song” (3m) “Hot Patootie - Bless My Soul” (3m) “I Cannot Be Played On Record Player 1” (2m) Each song is listed with its duration (in minutes). After one song finishes, the next (if any) immediately begins. For example, some possible concerts which last (exactly) 7 minutes: “Faerie’s Aire”, “Three Minute Song”, 0 minutes of reflection “Three Minute Song”, “Faerie’s Aire”, 0 minutes of reflection “Duetto Boffo”, “I Cannot Be Played On Record Player 1”, 1 minute of reflection “Hot Patootie”, “Hot Patootie”, 1 minute of reflection “I Cannot Be Played On Record Player 1”, 5 minutes of reflection 7 minutes of reflection Find a recurrence and appropriate initial conditions for the number of concerts which last (exactly) n minutes, n ≥ 0. Example of recurrence: an = an-1 + 3 * an-2 Where the initials are: a0 = 3 and a1 = 4 Give a brief justification why your recurrence is correct. (I’m not looking for a proof, but you do need to explain yourself.) (no justification = no credit) For your initial conditions, use as many as you need, but no more.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
As the leader of a popular indie math rock group, you would like to hold a concert. A
concert consists of a number of songs played in sequence, followed by a period of reflection.
The reflection lasts 0 or more minutes, in which the musicians sit on stage staring at the
audience, who are invited to meditate on the proceedings.
You have the following songs available:
“Faerie’s Aire and Death Waltz” (4m)
“Duetto Boffo di Due Gatti” (4m)
“Three Minute Song” (3m)
“Hot Patootie - Bless My Soul” (3m)
“I Cannot Be Played On Record Player 1” (2m)
Each song is listed with its duration (in minutes). After one song finishes, the next (if
any) immediately begins.
For example, some possible concerts which last (exactly) 7 minutes:
“Faerie’s Aire”, “Three Minute Song”, 0 minutes of reflection
“Three Minute Song”, “Faerie’s Aire”, 0 minutes of reflection
“Duetto Boffo”, “I Cannot Be Played On Record Player 1”, 1 minute of reflection
“Hot Patootie”, “Hot Patootie”, 1 minute of reflection
“I Cannot Be Played On Record Player 1”, 5 minutes of reflection
7 minutes of reflection
Find a recurrence and appropriate initial conditions for the number of concerts which last
(exactly) n minutes, n ≥ 0.
Example of recurrence: an = an-1 + 3 * an-2 Where the initials are: a0 = 3 and a1 = 4
Give a brief justification why your recurrence is correct.
(I’m not looking for a proof, but you do need to explain yourself.)
(no justification = no credit)
For your initial conditions, use as many as you need, but no more.
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