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For the following exercises, find the distinct number of arrangements. The letters in the word 鈥渁cademia鈥�

For the following exercises, find the distinct number of arrangements. The letters in the word 鈥渁cademia鈥� that begin and end in 鈥渁鈥�

, \\#, \\#, @, @, \$, \$, \$, \%, \%, 0, \%, \%\( # For the following exercises, find the distinct number of arrangements. The symbols in the string \)\\#, \\#, \\#, @, @, \$, \$, \$, \%, \%, 0, \%, \%$

, \\#, \\#, @, @, \$, \$, \$, \%, \%, \%, \%, \%\( that begin and end w鈥� # For the following exercises, find the distinct number of arrangements. The symbols in the string \)\\#, \\#, \\#, @, @, \$, \$, \$, \%, \%, \%, \%, \%\( that begin and end with 鈥淺)\%$鈥�

For the following exercises, find the distinct number of arrangements. The number of 5-element subsets from a set containing n elements is equal to the number of 6-element subsets from the same set. What is the value of n? (Hint: the order in which the elements for the subsets are chosen is not important.)

For the following exercises, find the distinct number of arrangements. \(\operatorname{Can} C(n, r)\) ever equal \(P(n, r) ?\) Explain.

For the following exercises, find the distinct number of arrangements. Suppose a set A has 2,048 subsets. How many distinct objects are contained in A?

For the following exercises, find the distinct number of arrangements. How many arrangements can be made from the letters of the word 鈥渕ountains鈥� if all the vowels must form a string?

A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back. a. How many arrangements are possible with no restrictions? b. How many arrangements are possible if the parents must sit in the front? c. How many arrangements are possible if the parents must be next to each other?

A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?