Question #348799

From a committee of 10 people,


1. In how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position?



2. In how many ways can we choose a subcommittee of 3 people?



3. Find the number of combinations of 13 objects taken 8 at a time.



4. How many 5-card hands will have 3 aces and 2 kings?



5. Serial numbers for a product are to be made using 2 letters followed by 3 numbers. If the letters are to be taken from the first 8 letters of the alphabet with no repeats and the numbers are taken from the 10 digits (0-9) with no repeats, how many serial numbers are possible?



6. A company has 7 senior and 5 junior officers. An ad hoc legislative committee is to be formed. In how many ways can a 4-officer committee be formed so that it is composed of



a. Any 4 officers?



b. 4 senior officers?



c. 3 senior officers and 1 junior officer?



d. 2 senior officers and 2 junior officers?



e. At least 2 senior officers?

1
Expert's answer
2022-06-08T13:18:09-0400

1.

10(9)(8)=72010(9)(8)=720

2.


(103)=120\dbinom{10}{3}=120

3.


(138)=1287\dbinom{13}{8}=1287

4.


(43)(42)=4(6)=24\dbinom{4}{3}\dbinom{4}{2}=4(6)=24

5.


8!(82)!10!(103)!=8(7)(10)(9)(8)=40320\dfrac{8!}{(8-2)!}\cdot \dfrac{10!}{(10-3)!}=8(7)(10)(9)(8)=40320

6.

a.

(7+54)=495\dbinom{7+5}{4}=495

b.

(74)(50)=35(1)=35\dbinom{7}{4}\dbinom{5}{0}=35(1)=35


c.

(73)(51)=35(5)=175\dbinom{7}{3}\dbinom{5}{1}=35(5)=175

d.

(72)(52)=21(10)=210\dbinom{7}{2}\dbinom{5}{2}=21(10)=210

e.

(72)(52)+(73)(51)+(74)(50)\dbinom{7}{2}\dbinom{5}{2}+\dbinom{7}{3}\dbinom{5}{1}+\dbinom{7}{4}\dbinom{5}{0}

=21(10)+35(5)+35(1)=420=21(10)+35(5)+35(1)=420


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