Answer :
All the solutions by applying the combination formula and binomial formula,
1. (11 9) = 55
2. (7 2) = 21
3. (10 5) = 252
4. The expansion of (x - 1)⁶ = x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1
5. The expansion of (x + 7)⁵ = x⁵ + 35x⁴ + 490x³ + 3430x² + 112005x + 16807
6. The expansion of (4x + 3)³ = 64x³ + 144x² + 108x + 27
7. the coefficient of x is 54.
8. the coefficient of x is 9375.
9. the coefficient of x is 19440.
Now simplify each part as;
1. (11 9)
It is equivalent to
= (11 2)
Since (n r) = n!/ r! (n - r)!
Hence we get;
11! / 2! 9!
55
Hence option D is true.
2. (7 2)
= 7!/2! (7 - 2)!
= 7!/2! 5!
= 21
Hence option C is true.
3. (10 5)
= 10! / 5! (10 - 5)!
= 10!/5! 5!
= 252
Hence option B is true.
4. The expansion of (x - 1)⁶;
Using binomial expansion;
(x - y)ⁿ = xⁿ - ⁿC₁ xⁿ⁻¹ (y) + ⁿC₂ xⁿ⁻² y² - ......... + (- 1)ⁿ yⁿ
(x - 1)⁶ = x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1
5. The expansion of (x + 7)⁵;
Using binomial expansion;
(x - y)ⁿ = xⁿ - ⁿC₁ xⁿ⁻¹ (y) + ⁿC₂ xⁿ⁻² y² - ......... + (- 1)ⁿ yⁿ
(x + 7)⁵ = x⁵ + 35x⁴ + 490x³ + 3430x² + 112005x + 16807
6. The expansion of (4x + 3)³;
Using binomial expansion;
(x - y)ⁿ = xⁿ - ⁿC₁ xⁿ⁻¹ (y) + ⁿC₂ xⁿ⁻² y² - ......... + (- 1)ⁿ yⁿ
(4x + 3)³ = 64x³ + 144x² + 108x + 27
7. Since we know that;
[tex]T_ {r + 1} = ^n C_ r (x)^{n - r} a^r[/tex]
Hence we get;
[tex]T_ {r + 1} = ^3 C_ r (2x)^{3 - r} 3^r[/tex]
For the coefficient of x;
3 - r = 1
r = 3 - 1
r = 2
Hence the coefficient of x is,
[tex]T_ {2 + 1} = ^3 C_ 2 (2x)^{3 - 2} 3^2[/tex]
[tex]T_ {2 + 1} = 54x[/tex]
Hence, option A is true.,
8. Since we know that;
[tex]T_ {r + 1} = ^n C_ r (x)^{n - r} a^r[/tex]
Hence we get;
[tex]T_ {r + 1} = ^5 C_ r (3x)^{5 - r} 5^r[/tex]
For the coefficient of x;
5 - r = 1
5 - 1 = r
r = 4
Hence the coefficient of x is,
[tex]T_ {4 + 1} = ^5 C_ 4 (3x)^{5 - 4} 5^4[/tex]
[tex]T_ {4 + 1} = 9375x[/tex]
Hence, option D is true.
9. Since we know that;
[tex]T_ {r + 1} = ^n C_ r (x)^{n - r} a^r[/tex]
Hence we get;
[tex]T_ {r + 1} = ^6 C_ r (3x)^{6 - r} 4^r[/tex]
For the coefficient of x⁴;
6 - r = 4
r = 2
Hence ;
[tex]T_ {r + 1} = ^6 C_ 2 (3x)^{6 - 2} 4^2[/tex]
[tex]T_ {2 + 1} = 19440[/tex]
Hence option C is true.
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