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Combinatorics, EXT1 A1 2022 HSC 11c

Find the coefficients of `x^(2)` and `x^(3)` in the expansion of `(1-(x)/(2))^(8)`.   (2 marks)

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`x^2: 7, \ x^3:-7`

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`text{General Term:}\ (1-(x)/(2))^(8) `

`T_k=((8),(k))(1)^(8-k)(-x/2)^k=((8),(k))(-1/2)^k x^k`

`text{Coefficient of}\ x^2 = ((8),(2))(-1/2)^2=7`

`text{Coefficient of}\ x^3 = ((8),(3))(-1/2)^3=-7`

Combinatorics, EXT1 A1 2019 13b

In the expansion of  `(5x + 2)^20`, the coefficients of `x^r` and `x^(r + 1)`  are equal.

What is the value of `r` ?   (3 marks)

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`14`

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`(5x + 2)^20`

COMMENT: Arithmetic becomes easier by expanding `(2 + 5x)^20`.

`text(General term:)\ (2 + 5x)^20`

`T_r` `= \ ^20C_r · 2^(20-r) · (5x)^r= \ ^20C_r · 2^(20-r) · 5^r · x^r`
`T_(r + 1)` `=\ ^20C_(r + 1) · 2^(19-r) · 5^(r + 1) · x^(r + 1)`

 
`text(Equating co-efficients:)`

`(20!)/(r!(20-r)!) · 2^(20-r) · 5^r` `= (20!)/((r + 1)!(19-r)!) · 2^(19-r) · 5 ^(r + 1)`
`2/(20-r)` `= 5/(r + 1)`
`2r + 2` `= 100-5r`
`7r` `= 98`
`r` `= 14`

Combinatorics, EXT1 A1 EQ-Bank 18

Find the coefficient of `x^4` in the expansion of  `(x^2-3/x)^5`.   (2 marks)

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`90`

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`text(General term:)`

`T_k` `= \ ^5C_k(x^2)^(5-k) · (-3/x)^k`
  `= \ ^5C_k · x^(10-2k)(-3)^k · x^(-k)`
  `= \ ^5C_k · x^(10-3k) · (-3)^k`

 
`text(Coefficient of)\ x^4\ text(occurs when)`

`10-3k= 4\ \ =>\ \ k=2`

`:.\ text(Coefficient of)\ \ x^4 = \ ^5C_2·(-3)^2 = 90`

Combinatorics, EXT1 A1 2015 HSC 13b

Consider the binomial expansion

`(2x + 1/(3x))^18 = a_0x^(18) + a_1x^(16) + a_2x^(14) + …`

where `a_0, a_1, a_2`, . . . are constants.

  1. Find an expression for `a_2`.  (2 marks)

  2. Find an expression for the term independent of `x`.  (2 marks)

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a.    `(\ ^18C_2 · 2^(16))/(3^2)`

b.    `(\ ^(18)C_9 · 2^9)/(3^9)`

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a.     `text(Need co-efficient of)\ x^(14)`

`text(General term of)\ (2x + 1/(3x))^(18)`

`T_k` `= \ ^(18)C_k(2x)^(18-k) · (1/(3x))^k`
  `= \ ^(18)C_k · 2^(18-k) · x^(18-k) · 3^(-k) · x^(-k)`
  `= \ ^(18)C_k · 2^(18-k) · 3^(-k) · x^(18-2k)`

 
`a_2\ text(occurs when:)`

`18-2k= 14\ \ =>\ \ k=2`

`:.a_2= \ ^(18)C_2 · 2^(18-2) · 3^(-2)= (\ ^(18)C_2 · 2^(16))/(3^2)`
 

b.    `text(Independent term occurs when:)`

`18-2k= 0\ \ =>\ \ k=9`

`:.\ text(Independent term)= \ ^(18)C_9 · 2^(18-9) · 3^(-9) = (\ ^(18)C_9 · 2^9)/(3^9)`

Combinatorics, EXT1 A1 2008 HSC 1d

Find an expression for the coefficient of `x^8 y^4` in the expansion of `(2x + 3y)^12`.   (2 marks) 

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`10\ 264\ 320`

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`text(Find co-efficient of)\ x^8 y^4 :`

MARKER’S COMMENT: More errors were made by students who used `T_(k+1)` as the general term rather than `T_k` (both are possible).

`T_k =\ text(General term of)\ (2x + 3y)^12 `

`T_k` `= ((12),(k)) (2x)^(12-k) * (3y)^k`
  `= ((12),(k)) * 2^(12-k) * 3^k * x^(12-k) * y^k`

 
`x^8 y^4\ text(occurs when)\ k = 4`

`T_4= ((12),(4)) * 2^(12-4) * 3^4 * x^8 y^4`

`:.\ text(Co-efficient of)\ x^8y^4= ((12),(4)) * 2^8 * 3^4= 10\ 264\ 320`

Combinatorics, EXT1 A1 2011 HSC 2c

Find an expression for the coefficient of `x^2` in the expansion of `(3x-4/x)^8`.   (2 marks)

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 `-870\ 912`

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`(3x-4/x)^8 = sum_(k=0)^8\ ^8C_k *(3x)^(8-k) *(-4/x)^k`

`text(General term)` `=\ ^8C_k * 3^(8-k) * x^(8-k) * (-1)^k*4^k * x^(-k)`
  `=\ ^8C_k*(-1)^k * 3^(8-k) * 4^k * x^(8-2k)`

 
`text(Co-efficient of)\ x^2=2\ text(occurs when:)`

`8-2k=2\ \ =>\ \ k=3`

`:.\ text(Co-efficient of)\ x^2`

`=\ ^8C_3 * (-1)^3*3^5 * 4^3`

`= -56 xx 243 xx 64`

`= -870\ 912`