Attempt 1 failed: You have exhausted your capacity on this model. Your quota will reset after 0s.. Retrying after 471.280936ms... Key Concept: General Term in Binomial Expansion

For a binomial expansion of the form $(a+b)^n$, the general term (or $(r+1)^{th}$ term) is given by the formula: $T_{r+1} = {}^{n}C_r a^{n-r} b^r$ where $n$ is a positive integer, $r$ is an integer such that $0 \le r \le n$, and ${}^{n}C_r = \frac{n!}{r!(n-r)!}$ is the binomial coefficient.

For a term $T_{r+1}$ to be an integral term, two conditions must be met:

1. The binomial coefficient ${}^{n}C_r$ is always an integer for $0 \le r \le n$.
2. The powers of $a$ and $b$, i.e., $a^{n-r}$ and $b^r$, must result in integer values when combined with any integer coefficients. Specifically, if $a$ and $b$ involve roots or fractional exponents, these must resolve to integers. When the bases are prime numbers, their exponents in the simplified form must be non-negative integers for the overall term to be an integer.

Problem Statement

We need to find the number of integral terms in the expansion of $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$.

Step-by-Step Working

1. Write the General Term ($T_{r+1}$)

Let $a = 7^{1/2}$, $b = 11^{1/6}$, and $n = 824$. Applying the general term formula $T_{r+1} = {}^{n}C_r a^{n-r} b^r$: $T_{r+1} = {}^{824}C_r \left(7^{1/2}\right)^{824-r} \left(11^{1/6}\right)^r$ $T_{r+1} = {}^{824}C_r (7)^{\frac{824-r}{2}} (11)^{\frac{r}{6}}$ Explanation: This step directly applies the binomial theorem's general term formula by substituting the base terms ($7^{1/2}$ and $11^{1/6}$) and the exponent of the binomial ($824$) into the formula. The powers are then simplified using exponent rules $(x^m)^n = x^{mn}$.

2. Establish Conditions for Integral Terms

For $T_{r+1}$ to be an integral term, the exponents of the prime numbers 7 and 11 must both be non-negative integers. This is because if a prime number is raised to a non-integer rational power (e.g., $7^{3/2}$), the result will not be an integer.

• Condition 1: Exponent of 7 must be an integer The exponent $\frac{824-r}{2}$ must be an integer. This implies that $824-r$ must be an even number. Since $824$ is an even number, for $824-r$ to be even, $r$ must also be an even number (an even number minus an even number equals an even number). So, $r$ must be a multiple of 2.

• Condition 2: Exponent of 11 must be an integer The exponent $\frac{r}{6}$ must be an integer. This implies that $r$ must be a multiple of 6.

Explanation: We analyze each exponent to determine the conditions on $r$. Since 7 and 11 are prime numbers, for $7^P \cdot 11^Q$ to be an integer, $P$ and $Q$ themselves must be integers. This leads to divisibility conditions for $r$.

3. Combine Conditions and Determine Possible Values of $r$

For $T_{r+1}$ to be an integral term, both Condition 1 and Condition 2 must be satisfied:

• $r$ must be a multiple of 2.
• $r$ must be a multiple of 6.

For both conditions to hold simultaneously, $r$ must be a common multiple of 2 and 6. The least common multiple (LCM) of 2 and 6 is 6. Therefore, $r$ must be a multiple of 6. It is important to note that if $r$ is a multiple of 6, it is automatically a multiple of 2, so the condition "$r$ is a multiple of 6" is sufficient.

Additionally, $r$ is an index in the binomial expansion, so it must satisfy $0 \le r \le n$. In this problem, $n=824$, so $0 \le r \le 824$.

Combining these, we are looking for values of $r$ such that:

1. $r$ is a multiple of 6.
2. $0 \le r \le 824$.

The possible values for $r$ are $0, 6, 12, 18, \dots$. To find the largest multiple of 6 that does not exceed 824, we divide 824 by 6: $824 \div 6 = 137$ with a remainder of 2. So, the largest multiple of 6 less than or equal to 824 is $6 \times 137 = 822$.

Thus, the sequence of possible values for $r$ is $0, 6, 12, \dots, 822$. Explanation: We find the LCM of the denominators of $r$ from the exponent conditions. Then, we apply the fundamental constraint that the index $r$ in a binomial expansion must be between 0 and $n$ (inclusive). This gives us an arithmetic progression of valid $r$ values.

4. Count the Number of Integral Terms

The values of $r$ form an arithmetic progression: $0, 6, 12, \dots, 822$. To count the number of terms in this sequence, we can use the formula: Number of terms = $\frac{\text{Last Term} - \text{First Term}}{\text{Common Difference}} + 1$ Number of terms = $\frac{822 - 0}{6} + 1$ Number of terms = $\frac{822}{6} + 1$ Number of terms = $137 + 1$ Number of terms = $138$

Alternatively, we can express $r$ as $6k$, where $k$ is an integer. From $0 \le r \le 824$, we have: $0 \le 6k \le 824$ Dividing by 6: $0 \le k \le \frac{824}{6}$ $0 \le k \le 137.33\dots$ Since $k$ must be an integer, the possible values for $k$ are $0, 1, 2, \dots, 137$. The number of such integer values for $k$ is $137 - 0 + 1 = 138$.

Therefore, there are 138 integral terms in the expansion.

Tips and Common Mistakes

• Range of $r$: Always remember that the index $r$ in the general term $T_{r+1}$ of a binomial expansion $(a+b)^n$ must satisfy $0 \le r \le n$. This constraint is vital for correctly identifying the bounds for $r$.
• Conditions for Integral Exponents with Prime Bases: When the base of an exponential term is a prime number (like 7 or 11 in this case) raised to a fractional power, for the result to be an integer, the exponent itself must resolve to an integer. Do not confuse this with situations where the base is a composite number that is a perfect power (e.g., $4^{1/2} = (2^2)^{1/2} = 2^1 = 2$, where the exponent $1/2$ is not an integer, but the base is not prime).
• Least Common Multiple (LCM): If $r$ needs to satisfy multiple divisibility conditions (e.g., $r$ must be a multiple of $k_1$ AND a multiple of $k_2$), then $r$ must be a multiple of the LCM of $k_1$ and $k_2$. Satisfying just one condition is insufficient.

Summary and Key Takeaway

To determine the number of integral terms in a binomial expansion involving terms with fractional exponents (like $(X^{1/p} + Y^{1/q})^n$):

1. Formulate the general term $T_{r+1}$.
2. Identify the exponents of the base terms ($X$ and $Y$).
3. Establish conditions on $r$ such that these exponents are non-negative integers (especially critical when $X$ and $Y$ are prime numbers).
4. Find the LCM of the denominators involved in the conditions for $r$. This LCM determines the required divisibility of $r$.
5. Identify the valid range for $r$ using $0 \le r \le n$.
6. Count all multiples of the LCM that fall within this valid range of $r$. This count gives the number of integral terms.