Just going beyond Binomials

Probability Level 5

Find the coefficient of x 10 x^{10} in the expansion of ( 1 − 2 x + 3 x 2 − 4 x 3 + 5 x 4 − 6 x 5 + 7 x 6 ) 7 \large (1 - 2x + 3x^{2} - 4x^{3} + 5x^{4} - 6x^{5} +7x^{6})^{7} .


The answer is 1117851.

Mohit Gupta by Mohit Gupta , 20, India

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1 solution

Nikola Djuric
Feb 29, 2016

g ( x ) = f ′ ( x ) = 1 − 2 x + 3 x 2 − 4 x 3 + 5 x 4 − 6 x 5 + 7 x 6 g(x)=f'(x)=1-2x+3x^2-4x^3+5x^4-6x^5+7x^6

f ( x ) = x − x 2 + x 3 − x 4 + x 5 − x 6 + x 7 = x ( 1 + x 7 ) / ( 1 + x ) = ( x 8 + x ) / ( x + 1 ) f(x)=x-x^2+x^3-x^4+x^5-x^6+x^7=x(1+x^7)/(1+x)=(x^8+x)/(x+1)

g ( x ) = f ′ ( x ) = ( 7 x 8 + 8 x 7 + 1 ) / ( x + 1 ) 2 g(x)=f'(x)=(7x^8+8x^7+1)/(x+1)^2

g ( x ) 7 = ( 823543 x 5 6 + 6588344 x 5 5 + 22588608 x 5 4 + 43025920 x 5 3 + 49172480 x 5 2 + 33718272 x 5 1 + 12845056 x 5 0 + 2097152 x 4 9 + 823543 x 4 8 + 5647152 x 4 7 + 16134720 x 4 6 + 24586240 x 4 5 + 21073920 x 4 4 + 9633792 x 4 3 + 1835008 x 4 2 + 352947 x 4 0 + 2016840 x 3 9 + 4609920 x 3 8 + 5268480 x 3 7 + 3010560 x 3 6 + 688128 x 3 5 + 84035 x 3 2 + 384160 x 3 1 + 658560 x 3 0 + 501760 x 2 9 + 143360 x 2 8 + 12005 x 2 4 + 41160 x 2 3 + 47040 x 2 2 + 17920 x 2 1 + 1029 x 1 6 + 2352 x 1 5 + 1344 x 1 4 + 49 x 8 + 56 x 7 + 1 ) / ( x 1 4 + 14 x 1 3 + 91 x 1 2 + 364 x 1 1 + 1001 x 1 0 + 2002 x 9 + 3003 x 8 + 3432 x 7 + 3003 x 6 + 2002 x 5 + 1001 x 4 + 364 x 3 + 91 x 2 + 14 x + 1 ) g(x)^7=(823543 x^56+6588344 x^55+22588608 x^54+43025920 x^53+49172480 x^52+33718272 x^51+12845056 x^50+2097152 x^49+823543 x^48+5647152 x^47+16134720 x^46+24586240 x^45+21073920 x^44+9633792 x^43+1835008 x^42+352947 x^40+2016840 x^39+4609920 x^38+5268480 x^37+3010560 x^36+688128 x^35+84035 x^32+384160 x^31+658560 x^30+501760 x^29+143360 x^28+12005 x^24+41160 x^23+47040 x^22+17920 x^21+1029 x^16+2352 x^15+1344 x^14+49 x^8+56 x^7+1)/(x^14+14 x^13+91 x^12+364 x^11+1001 x^10+2002 x^9+3003 x^8+3432 x^7+3003 x^6+2002 x^5+1001 x^4+364 x^3+91 x^2+14 x+1)

x 1 4 + 14 x 1 3 + 91 x 1 2 + 364 x 1 1 + 1001 x 1 0 + 2002 x 9 + 3003 x 8 + 3432 x 7 + 3003 x 6 + 2002 x 5 + 1001 x 4 + 364 x 3 + 91 x 2 + 14 x + 1 = 823543 x 4 2 − 4941258 x 4 1 + 16823807 x 4 0 − 42622552 x 3 9 + 89173140 x 3 8 − 162453032 x 3 7 + 265708772 x 3 6 − 397511568 x 3 5 + 550832261 x 3 4 − 713503350 x 3 3 + 869944509 x 3 2 − 1003726752 x 3 1 + 1100401288 x 3 0 − 1149988560 x 2 9 + 1148581320 x 2 8 − 1098636896 x 2 7 + 1008048510 x 2 6 − 888369300 x 2 5 + 752671374 x 2 4 − 613504080 x 2 3 + 481312048 x 2 2 − 363519328 x 2 1 + 264310032 x 2 0 − 184961728 x 1 9 + 124517890 x 1 8 − 80585820 x 1 7 + 50088626 x 1 6 − 29861664 x 1 5 + 17047992 x 1 4 − 9301040 x 1 3 + 4837112 x 1 2 − 2390304 x 1 1 + 1117851 x 1 0 − 492226 x 9 + 202755 x 8 − 77464 x 7 + 27132 x 6 − 8568 x 5 + 2380 x 4 − 560 x 3 + 105 x 2 − 14 x + 1 x^14+14 x^13+91 x^12+364 x^11+1001 x^10+2002 x^9+3003 x^8+3432 x^7+3003 x^6+2002 x^5+1001 x^4 +364 x^3+91 x^2+14 x+1=823543 x^42-4941258 x^41+16823807 x^40-42622552 x^39+89173140 x^38-162453032 x^37+265708772 x^36-397511568 x^35+550832261 x^34-713503350 x^33+869944509 x^32-1003726752 x^31+1100401288 x^30-1149988560 x^29+1148581320 x^28-1098636896 x^27+1008048510 x^26-888369300 x^25+752671374 x^24-613504080 x^23+481312048 x^22-363519328 x^21+264310032 x^20-184961728 x^19+124517890 x^18-80585820 x^17+50088626 x^16-29861664 x^15+17047992 x^14-9301040 x^13+4837112 x^12-2390304 x^11+1117851 x^10-492226 x^9+202755 x^8-77464 x^7+27132 x^6-8568 x^5+2380 x^4-560 x^3+105 x^2-14 x+1

so coefficient is 1117851

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Also 10=6+1x4=6+4x1=6+2x2=6+2+1x2=6+4x1=6+3+1=5x2=5+4+1=5+3+2=5+2x2+1=5+2+1x3=5+1x5=

4x2+2x1=4x2+1x2=4+3x2=4+3x1+1x3=4+2x3=4+2x2+1x2=4+2x1+1x4=4+1x6=3x3+1=3x2+2x2=

3x2+2x1+1x2=3x2+1x4=3+2x3+1=3+2x2+1x3=3+2+1x5=2x3+1x4

so coefficient is 7!/(4!) 7 (-2)^4 + 7!/(2! 2!) 7 3^2+ ... +7!/(3! 4!) 3^3 (-2)^4=1117851

Nikola Djuric - 5 years, 3 months ago

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Can you add words to explain what you're doing?

Note that you can edit your solution directly.

Calvin Lin Staff - 5 years, 3 months ago

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Could you please edit the solution in LATEX format.

Ahmed K - 5 years, 3 months ago

Are you serious? Thats not a solution....

Raunak Agrawal - 4 years, 3 months ago

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