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If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x

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div10
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If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 07:42
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If \((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\). What is the value of x+y+z+t?

A. 7
B. 15
C. 20
D. 30
E. 35
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D
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pushpitkc
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If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 08:25
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div10 wrote:
If \((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\). What is the value of x+y+z+t?

A. 7
B. 15
C. 20
D. 30
E. 35


We can solve the problem by using the pascals triangle - The coefficients(the numbers in front of each term) follow a pattern

1
\((a+b)^0 = 1\)

1 1
\((a+b)^1 = a + b\)

1 2 1
\((a+b)^2 =a^2 + 2ab +b^2\)

1 3 3 1
\((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

1 4 6 4 1
\((a + b)4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b4\)

1 5 10 10 5 1
\((a+b)^5 = a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5\)


Therefore, Value of x+y+z+t = \(5+10+10+5 = 30\)(Option D)
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chetan2u
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 09:23
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div10 wrote:
If \((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\). What is the value of x+y+z+t?

A. 7
B. 15
C. 20
D. 30
E. 35



Hi...

any expansion is ..
\((a+b)^n = nC0a^nb+nC1a^{n-1}b^1........+nCnb^n\)
and when you add the coefficient \(nC0+nC1+...nCn=2^n\)

here n is 5..
\((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\)
so addition of coefficients = \(2^5=1+x+y+z+t+1......32=2+x+y+z+t\)
so \(x+y+z+t =32-2=30\)

D
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saswata4s
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 09:43
chetan2u wrote:
div10 wrote:
If \((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\). What is the value of x+y+z+t?

A. 7
B. 15
C. 20
D. 30
E. 35



Hi...

any expansion is ..
\((a+b)^n = nC0a^nb+nC1a^{n-1}b^1........+nCnb^n\)
and when you add the coefficient \(nC0+nC1+...nCn=2^n\)


Just wondering if this is in scope in GMAT?
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chetan2u
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 09:45
Expert Reply
saswata4s wrote:
chetan2u wrote:
div10 wrote:
If \((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\). What is the value of x+y+z+t?

A. 7
B. 15
C. 20
D. 30
E. 35



Hi...

any expansion is ..
\((a+b)^n = nC0a^nb+nC1a^{n-1}b^1........+nCnb^n\)
and when you add the coefficient \(nC0+nC1+...nCn=2^n\)


Just wondering if this is in scope in GMAT?


No, not really..
But \(nC0+nC1+nC2+...nC(n-1)+nCn= 2^n\) can make some Q easier although there would be some other way too to solve it.
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 09:46
div10 wrote:
If \((a+b)^5=a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5\). What is the value of x+y+z+t?

A. 7
B. 15
C. 20
D. 30
E. 35


Binomial theorem has never been asked in the GMAT. The question per-say is not difficult if one knows the binomial theorem or expansion till power 5

Hi Bunuel, is there any other way to solve the problem?
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saswata4s
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   25 Oct 2017, 09:54
chetan2u wrote:
No, not really..
But \(nC0+nC1+nC2+...nC(n-1)+nCn= 2^n\) can make some Q easier although there would be some other way too to solve it.


Hmm, I agree. But I almost forgot all the concepts of binomial series and theorem, so this question really scared me :roll:
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink] New post   04 Oct 2023, 19:44
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Re: If (a+b)^5= a^5+xa^4b+ya^3b^2+za^2b^3+tab^4+b^5 What is the value of x [#permalink]
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