Expert: Scott A Wilson - 9/29/2009

Question
QUESTION: Hi.

I am wondering if you can help me solve this problem:

the binomial coefficient of the the 4th term of (a+b )^n is the double of the coefficient of the 3rd term of (c+d)^n-1 what is n

my solution :



since only the coefficient are related  I ignored the variables

so nCr = 2 n-1Cr



the r in the first binomial is 3 since its the 4th term

and the r in the second is 2 since its the 3rd term

nC3=2 n-1C2



now I did something I am not that sure of I said

n!/3!(n-3)! = 2 (n-1)!/2!(n-1-2)!

n(n-1)!/3!(n-3)! = 2 (n-1)!/2!(n-3)!



n/3! =  2/2!



n=3!

n=6



Is that correct?

ANSWER: I don't think so.  I believe it is n = 8.
What you are looking for is a row in which 4th term is twice the 3rd.
For n = 0, 1, 2 don't have enough terms.

The 3rd row in the binomial triangle is 1 3 3 1.

The 4th row is 1 4 6 4 1.
Notice that each row is the sum of the two terms above the numbers.
1 = 1, 4=1+3, 6=3+3, 4=3+1, 1=1.

The 5th row is 1 5 10 10 5 1.

The 6th row is 1 6 15 20 15 6 1.

The 7th row is 1 7 21 35 35 21 7 1.

The 8th row is the one we're looking for, for it is
1 8 28 56 70 56 28 8 1 and it can be seen that 65 is twice 25.


---------- FOLLOW-UP ----------

QUESTION: but
6C3
6!/3!*3!
6*5*4/3*2
2*5*2
=20

5C2
5!/3!*2!
5*4/2
5*2=10
2*10=20...
also how is 65 twice 25?


ANSWER: The answer is in the 8th row.

8C3 = 8*7*6/3*2*1=56

8C2 = 8*7/2*1=56/2=28 (which is half of 56, as was just seen).


---------- FOLLOW-UP ----------

QUESTION: the binomial coefficient of the the 4th term of (a+b )^n is the double of the coefficient of the 3rd term of (c+d)^"""n-1""" what is n

why did you write 8 while the question says that the second binomial is powered to n-1
so it should be n-1Cr 7C2 =7*6 = 42 isnt half of 56

Answer
The 1st term in each row is nC0.
The 2nd term in each row is nC1.
The 3rd term in each row is nC2.
The 4th term in each row is nC3.

This says that n must be at least 4.
Try n=4: 4C3=4, 4C2=6 so 4C3/4C2 = 4/6 = 2/3.
Try n=5: 5C3=10, 5C2=10, so 5C3/5C2 is 10/10 = 1.
Try n=6: 6C3=20, 6C2=15, so 6C3/6C2 is 20/15 = 4/3.
Try n=7: 7C3=35, 7C2=21, so 7C3/7C2 is 35/21 = 5/3.
Try n=8: 8C3=56, 8C2=28, so 8C3/4C2 is 56/28 = 2.