Question

Show that the number of ways in which four distinct integers can be chosen from {
1
,
2
,
.
.
,
n
}
,
n
≥
7
such that no two are consecutive is equal to (
n
−
3
4
)

Answer 1

Answer:

n-3 C 4

Step-by-step explanation:  

Consider the n numbers: Now from the 4 no.s to be selected they are all non-consecutive and hence we have to remove 3 no.s in between. We will continue to remove numbers until we are left with 4. Now clearly there are n-7 numbers left. Think of it this way the O represents 4 no.s we keep and the | represents number we remove and their position represents the number:

          O|O|O|O

Now we use the formula for putting r objects which are the | in this case in the 5 available spaces between and to the side of O's which is

n+r-1Cr-1    Now n is n-7 and r is 5 hence the answer will be n-3 C 4