Question

In how many ways can 6 competitors finish a race assuming there are no ties?In how many ways can we sit n people on r < n chairs? In how many wayscan the gold, silver and bronze medals be awarded in a race of 8 people? Howmany 5-digit zip-codes are possible? How many different 3-person committeeswe can make with 10 people?

Answer 1

Answer: a) 720, b)
`^nC_r=(n!)/((n-r)!r!)` c) 336, d) 30240 e) 120.

Explanation:

Since we have given that

a) Number of competitors = 6

We need to find the number of ways that 6 competitors can finish a race and there are no ties.

So, Number of ways would be

`6!=720`

b) number of people = n

Number of chairs = r (r<n)

so, the number of ways we can sit n people on r<n chairs is given by

`^nC_r=(n!)/((n-r)!r!)`

c) If there are three awards :

Gold, silver, bronze.

We need to find the number of ways that these medals can be awarded.

Number of ways is given by

`^8P_3\\\\=336`

d) Number of 5-digit zip codes are possible is given by

`^(10)P_5=30240`

e) Number of poeple = 10

Number of people required in committee = 3

Number of different 3 person committee that are selected from 10 people is given by

`^(10)C_3=120`

Hence, a) 720, b)
`^nC_r=(n!)/((n-r)!r!)` c) 336, d) 30240 e) 120.