Question

In the story, 12 districts must regularly send a teen boy and girl known as "Tributes" to the Capitol district to compete in the hunger games. As the games begin, President Coriolaunus Snow addresses the contestants with a traditional phrase "May the odds be ever in your favor."

a) How many different first names could President Snow create using the letters in Coriolanus?
3906
b) How many different first names could he create, if the consonants in Coriolaúnus are kept together?
saal
06
c) How many different first names could he create using Coriolaunus, that end with an 'vowel?
d) How many different first names could he create, if he kept the consonants in alphabetical order?
e) How many different first names could he create, if he didn't repeat a letter in that first name?

Katniss Everdeen volunteered to become a "tribute" after her younger sister was selected for the hunger games. Assuming that there were 64 boys and 59 girls who were eligible to be selected that year from her district as "tributes," then:
a) how many ways could 2 youths be selected from the entire group of eligible youths?
b) how many ways could a single boy and girl be selected as "tributes" from the eligible youths?
c) Some families had to offer more than one child for the selection process. If there were 14 pairs of brother and sister groups among the eligible youths, then how many ways could a girl be selected first followed by her brother to compete in the hunger games?

Answer 1

a) To find the number of different first names President Snow could create using the letters in "Coriolanus", we can use the formula for permutations with repeated letters. There are 10 letters in "Coriolanus", but "o" and "i" each appear twice, so the total number of permutations is:

10! / (2! * 2!) = 45,360 / 4 = 11,340

Therefore, President Snow could create 11,340 different first names using the letters in "Coriolanus".

b) If the consonants in "Coriolanus" are kept together, we have "Crlns" as a string of consonants. This gives us 5 consonants to arrange, so the number of permutations is:

5! = 120

Therefore, President Snow could create 120 different first names using the consonants in "Coriolanus" kept together.

c) To count the number of different first names using "Coriolanus" that end with a vowel, we can consider the last letter of the name. There are 4 vowels in "Coriolanus", so there are 4 choices for the last letter. For the other letters, we can use the remaining 9 letters (excluding the last vowel) in any order. Therefore, the number of different first names that end with a vowel is:

4 * 9! = 1451520

Therefore, President Snow could create 1,451,520 different first names using "Coriolanus" that end with a vowel.

d) If the consonants in "Coriolanus" are kept in alphabetical order, then we have "aclnorsu". This gives us 8 letters to arrange, so the number of permutations is:

8! = 40,320

Therefore, President Snow could create 40,320 different first names using the consonants in "Coriolanus" in alphabetical order.

e) To find the number of different first names President Snow could create without repeating any letters, we can use the formula for permutations without repetition. There are 10 letters in "Coriolanus", so the total number of permutations is:

10! = 3,628,800

Therefore, President Snow could create 3,628,800 different first names without repeating any letters.

a) To find the number of ways to select 2 youths from the group of eligible youths, we can use the formula for combinations. We have 64 boys and 59 girls, so the total number of eligible youths is 64 + 59 = 123. The number of ways to select 2 youths is:

123C2 = (123 * 122) / 2 = 7503

Therefore, there are 7,503 ways to select 2 youths from the entire group of eligible youths.

b) To find the number of ways to select a single boy and girl as "tributes" from the eligible youths, we can use the product rule. There are 64 boys to choose from and 59 girls to choose from, so the number of ways to select one boy and one girl is:

64 * 59 = 3,776

Therefore, there are 3,776 ways to select a single boy and girl as "tributes" from the eligible youths.

c) To find the number of ways to select a girl first followed by her brother, we can use the product rule again. There are 59 girls to choose from for the first selection, and after a girl is selected, there are 63 youths left to choose from (excluding the selected girl and the 14 pairs of brother and sister groups). Therefore, the number of ways to select a girl first followed by her brother is:

59 *