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Sagot :
To determine which of the statements regarding the expansion of [tex]\((x+y)^n\)[/tex] are correct, let's carefully analyze each statement one by one:
### Statement A:
"The coefficients of [tex]\(x^n\)[/tex] and [tex]\(y^n\)[/tex] both equal 1."
In the expansion of [tex]\((x+y)^n\)[/tex]:
- The term [tex]\(x^n\)[/tex] corresponds to only one combination, where [tex]\(x\)[/tex] is chosen [tex]\(n\)[/tex] times and [tex]\(y\)[/tex] is not chosen.
- Likewise, the term [tex]\(y^n\)[/tex] corresponds to only one combination, where [tex]\(y\)[/tex] is chosen [tex]\(n\)[/tex] times and [tex]\(x\)[/tex] is not chosen.
Therefore, the coefficients of [tex]\(x^n\)[/tex] and [tex]\(y^n\)[/tex] are indeed both 1.
This statement is correct.
### Statement B:
"The coefficients of [tex]\(x^{n-1}\)[/tex] and [tex]\(y^{n-1}\)[/tex] both equal 1."
- In the expansion of [tex]\((x+y)^n\)[/tex], the term [tex]\(x^{n-1}\)[/tex] would be associated with the term [tex]\(\binom{n}{n-1} x^{n-1} y^1\)[/tex].
- Similarly, the term [tex]\(y^{n-1}\)[/tex] would be associated with the term [tex]\(\binom{n}{1} x^1 y^{n-1}\)[/tex].
The binomial coefficients are [tex]\(\binom{n}{n-1} = n\)[/tex] and [tex]\(\binom{n}{1} = n\)[/tex], so the coefficients of [tex]\(x^{n-1}\)[/tex] and [tex]\(y^{n-1}\)[/tex] are both [tex]\(n\)[/tex], not 1.
This statement is incorrect.
### Statement C:
"The coefficients of [tex]\(x^a y^b\)[/tex] and [tex]\(x^b y^a\)[/tex] are equal."
- In the binomial expansion [tex]\((x+y)^n\)[/tex], the general term is given by [tex]\(\binom{n}{a} x^a y^{n-a}\)[/tex], where [tex]\(a + b = n\)[/tex].
- For the term [tex]\(x^a y^b\)[/tex], the coefficient is [tex]\(\binom{n}{a}\)[/tex].
- For the term [tex]\(x^b y^a\)[/tex], the coefficient is [tex]\(\binom{n}{b}\)[/tex].
Since [tex]\(\binom{n}{a} = \binom{n}{b}\)[/tex] (by the symmetry property of the binomial coefficient), the coefficients are indeed equal.
This statement is correct.
### Statement D:
"For any term [tex]\(x^a y^b\)[/tex] in the expansion, [tex]\(a-b=n\)[/tex]."
In the binomial expansion [tex]\((x+y)^n\)[/tex],
- Each term is of the form [tex]\(x^a y^b\)[/tex], with [tex]\(a + b = n\)[/tex].
- The correct relationship is [tex]\(a + b = n\)[/tex].
The statement [tex]\(a - b = n\)[/tex] is not true, as it contradicts the fundamental relationship in the binomial expansion.
This statement is incorrect.
### Conclusion:
The correct statements are:
- A. The coefficients of [tex]\(x^n\)[/tex] and [tex]\(y^n\)[/tex] both equal 1.
- C. The coefficients of [tex]\(x^a y^b\)[/tex] and [tex]\(x^b y^a\)[/tex] are equal.
Thus, the correct statements regarding the expansion of [tex]\((x+y)^n\)[/tex] are A and C.
### Statement A:
"The coefficients of [tex]\(x^n\)[/tex] and [tex]\(y^n\)[/tex] both equal 1."
In the expansion of [tex]\((x+y)^n\)[/tex]:
- The term [tex]\(x^n\)[/tex] corresponds to only one combination, where [tex]\(x\)[/tex] is chosen [tex]\(n\)[/tex] times and [tex]\(y\)[/tex] is not chosen.
- Likewise, the term [tex]\(y^n\)[/tex] corresponds to only one combination, where [tex]\(y\)[/tex] is chosen [tex]\(n\)[/tex] times and [tex]\(x\)[/tex] is not chosen.
Therefore, the coefficients of [tex]\(x^n\)[/tex] and [tex]\(y^n\)[/tex] are indeed both 1.
This statement is correct.
### Statement B:
"The coefficients of [tex]\(x^{n-1}\)[/tex] and [tex]\(y^{n-1}\)[/tex] both equal 1."
- In the expansion of [tex]\((x+y)^n\)[/tex], the term [tex]\(x^{n-1}\)[/tex] would be associated with the term [tex]\(\binom{n}{n-1} x^{n-1} y^1\)[/tex].
- Similarly, the term [tex]\(y^{n-1}\)[/tex] would be associated with the term [tex]\(\binom{n}{1} x^1 y^{n-1}\)[/tex].
The binomial coefficients are [tex]\(\binom{n}{n-1} = n\)[/tex] and [tex]\(\binom{n}{1} = n\)[/tex], so the coefficients of [tex]\(x^{n-1}\)[/tex] and [tex]\(y^{n-1}\)[/tex] are both [tex]\(n\)[/tex], not 1.
This statement is incorrect.
### Statement C:
"The coefficients of [tex]\(x^a y^b\)[/tex] and [tex]\(x^b y^a\)[/tex] are equal."
- In the binomial expansion [tex]\((x+y)^n\)[/tex], the general term is given by [tex]\(\binom{n}{a} x^a y^{n-a}\)[/tex], where [tex]\(a + b = n\)[/tex].
- For the term [tex]\(x^a y^b\)[/tex], the coefficient is [tex]\(\binom{n}{a}\)[/tex].
- For the term [tex]\(x^b y^a\)[/tex], the coefficient is [tex]\(\binom{n}{b}\)[/tex].
Since [tex]\(\binom{n}{a} = \binom{n}{b}\)[/tex] (by the symmetry property of the binomial coefficient), the coefficients are indeed equal.
This statement is correct.
### Statement D:
"For any term [tex]\(x^a y^b\)[/tex] in the expansion, [tex]\(a-b=n\)[/tex]."
In the binomial expansion [tex]\((x+y)^n\)[/tex],
- Each term is of the form [tex]\(x^a y^b\)[/tex], with [tex]\(a + b = n\)[/tex].
- The correct relationship is [tex]\(a + b = n\)[/tex].
The statement [tex]\(a - b = n\)[/tex] is not true, as it contradicts the fundamental relationship in the binomial expansion.
This statement is incorrect.
### Conclusion:
The correct statements are:
- A. The coefficients of [tex]\(x^n\)[/tex] and [tex]\(y^n\)[/tex] both equal 1.
- C. The coefficients of [tex]\(x^a y^b\)[/tex] and [tex]\(x^b y^a\)[/tex] are equal.
Thus, the correct statements regarding the expansion of [tex]\((x+y)^n\)[/tex] are A and C.
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