Answer :
ANSWER
a) Use the homogeneous property of the binomial expansion to find the missing exponent
b) Use the binomial theorem to find the coefficient
EXPLANATION
The given binomial expansion is:
[tex](x+y)^{12} [/tex]
When we compare this to
[tex](a + b) ^{n} [/tex]
We have
[tex]n = 12[/tex]
Therefore the of each term in the expansion must be 12.
[tex] \implies \: 7 + b = 12[/tex]
[tex]b = 12 - 7[/tex]
[tex]b = 5[/tex]
Since the coefficient of x and y are unity, we use the formula
[tex]^{n} C_r = \frac{n!}{(n - r)!r!} [/tex]
to find the coefficient.
Where n=12 and r=5(the exponent of the y-term).
Therefore the coefficient is
[tex]^{12} C_5= \frac{12!}{(12- 5)!5!} [/tex]
[tex]^{12} C_5= \frac{12!}{7!5!} [/tex]
[tex]^{12} C_5= \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{7! \times 5 \times 4 \times 3 \times 2 \times 1} [/tex]
When we simplify further we get:
[tex]^{12} C_5= 11 \times 9 \times 8 = 792[/tex]