Answer: a. 120, b. 386, c. 176, d. 968.
Step-by-step explanation:
For a combination of any number, is given as
C n,r = n!/r!(n-r)!
Please note that "n,r" is a subscript, and the exclamation mark "!" is called factorial.
From the question, n = 10
EXACTLY 3 0s
10 combination 3
r is exactly 3, that is equal 3.
C 10,3= 10!/3!(10-3)! = 10!/3!7!= 120.
For clarification,
10!/3!7!=10×9×8/3×2×1 = 120.
You can also use a calculator to compute the factorials.
MORE 0s than 1s
There will be more 0s than 1s when < 5bits are 0s.
We have r<5
Therefore for r=4
C 10,4 = 10!/4!(10-4)!=10!/4!6!=210
r=3
C 10,3= 10!/3!(10-3)!=10!/3!7!=120
r=2
C 10,2=10!/2!(10-2)!=10!/2!8!=45
r=1
C 10,1=10!/1!(10-1)!=10!/1!9!=10
r=0
C 10,0=10!/0!(10-0)!=10!/0!10!=1
Summing the answers gives us our final answer
210+120+45+10+1= 386.
AT LEAST 7 1s
To get this combination, the value of r will be greater than or equal to 7
r>=7
We have,
r=7
C 10,7=10!/7!(10-7)!=10!/7!3!=120
r=8
C 10,8=10!/8!(10-8)!=10!/8!2!=45
r=9
C 10,9=10!/9!(10-9)!=10!/9!1!=10
r=10
C 10,10=10!/10(10-10)!=10!/10!0!=1
120+45+10+1= 176
AT LEAST 3 1s
the value for r will be greater than or equal to 3:
We can the values of r from 3 to 10.
r=3
C 10,3=10!/3!(10-3)!=120
r=4
C 10,4=10!/4!(10-4)!=10!/4!6!=210
r=5
C 10,5=10!/5!(10-5)!=10!/5!5!=252
r=6
C 10,6=10!/6!(10-6)!=10!/6!4!=210
r=7
C 10,7=10!/7!(10-7)!=10!/7!3!=120
r=8
C 10,8=10!/8!(10-8)!=10!/8!2!=45
r=9
C 10,9=10!/9!(10-9)!=10!/9!1!=10
r=10
C 10,10=10!/10!(10-10)!=10!/10!0!=1
Adding our answers gives 968.
