Counting principles and the extended binomial theorem

For a positive integer $n$, the factorial $n!$ is the product of all positive integers from $n$ down to 1. Factorials arise whenever distinct objects are placed into distinct positions.

By definition, $0!=1$. This convention makes permutation, combination, and binomial-coefficient formulas work correctly at their endpoints.

Factorials grow extremely quickly. Simplify ratios before evaluating: for example, $\frac{12!}{10!}=12\times11$, not two enormous calculator values.

Equations involving factorials often simplify by cancellation. Always state the integer restriction: factorial notation in this course normally requires $n\in\{0,1,2,\ldots\}$.

A permutation is an ordered arrangement. Changing the order creates a different outcome, so ABC and BAC are distinct permutations.

Arranging all $n$ distinct objects gives $n!$ outcomes. Arranging $r$ objects selected from $n$ distinct objects gives ${}_{n}P_r=\frac{n!}{(n-r)!}$.

The product rule explains the formula: there are n choices for the first position, n − 1 for the second, and so on until r positions are filled.

Use permutations for ranked places, assigned roles, ordered codes without repetition, or arrangements in a line.

A combination is an unordered selection. The group {A, B, C} is the same group as {C, A, B}, so different internal orders must not be counted again.

The number of ways to select $r$ objects from $n$ distinct objects is $\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Dividing by $r!$ removes the $r!$ orders of each selected group.

The symmetry identity $\binom{n}{r}=\binom{n}{n-r}$ reflects the equivalence of choosing the $r$ included objects or the $n-r$ excluded objects.

Use combinations for committees, teams, hands of cards, or any selection where the order of the chosen members has no meaning.

Restrictions change the sample space. Translate each condition before choosing a formula: ‘together’, ‘not together’, ‘at least’, ‘exactly’, ‘no repetition’, and ‘repetition allowed’ lead to different structures.

For objects that must stay together, treat the required group as one block, arrange the block with the remaining objects, then arrange the objects inside the block.

For objects that must not be together, the complement method is often efficient: count all arrangements and subtract those in which the forbidden objects form a block.

If $n$ positions contain groups of identical objects of sizes $n_1,n_2,\ldots,n_k$, the number of distinguishable arrangements is $\frac{n!}{n_1!n_2!\cdots n_k!}$, where $n_1+\cdots+n_k=n$.

Do not confuse identical objects with repetition allowed. A length-$r$ code chosen from $k$ symbols with reuse allowed has $k^r$ outcomes; arranging a fixed multiset requires division by repeated factorials.

The coefficient $\binom{n}{r}$ counts the ways to choose the $r$ positions that receive the second component $b$ when expanding $(a+b)^n$. This connects combinations directly to algebra.

For $n\in\{0,1,2,\ldots\}$, the finite binomial theorem is $(a+b)^n=\sum\limits_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$. It contains exactly $n+1$ terms.

The general term is $T_{r+1}=\binom{n}{r}a^{n-r}b^r$. The index starts at $r=0$, so the term containing $r$ is term number $r+1$.

If the second component is negative, include its sign inside the power. This automatically produces the correct alternating sign pattern.

The extended, or generalized, binomial theorem allows an exponent $\alpha$ that is negative or non-integer. Instead of stopping, the expansion usually continues indefinitely.

For $|x|<1$, $(1+x)^\alpha=\sum\limits_{r=0}^{\infty}\binom{\alpha}{r}x^r$, where $\binom{\alpha}{r}=\frac{\alpha(\alpha-1)\cdots(\alpha-r+1)}{r!}$ and $\binom{\alpha}{0}=1$.

The first terms are $1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3+\cdots$. This form is often the fastest way to generate the required number of terms.

If $\alpha=n$ is a non-negative integer, the factor $n-n=0$ makes all later coefficients vanish, so the generalized formula reduces to the familiar finite theorem.

Negative powers often generate recognizable infinite patterns. For example, $(1+x)^{-1}=1-x+x^2-x^3+\cdots$, which is also an infinite geometric series.

For a positive integer $m$, $\binom{-m}{r}=(-1)^r\binom{m+r-1}{r}$. This identity explains the alternating generalized coefficients before any substitution inside $(1+x)^{-m}$.

Fractional powers model roots. The expansion of $(1+x)^{\frac{1}{2}}$ can approximate $\sqrt{1+x}$ when $|x|<1$, and $(1+x)^{\frac{-1}{2}}$ can approximate $\frac{1}{\sqrt{1+x}}$.

For $(a+bx)^\alpha$, factor first: $(a+bx)^\alpha=a^\alpha\left(1+\frac{b}{a}x\right)^\alpha$. The convergence condition becomes $\left|\frac{b}{a}x\right|<1$.

For the extended expansion $(1+u)^\alpha$, the general term is $T_{r+1}=\binom{\alpha}{r}u^r$. It works for any non-negative integer $r$, even though $\alpha$ need not be an integer.

To find a coefficient, express u in terms of x, identify the value of r that produces the required power, and include every numerical factor from u raised to that power.

For a transformed expression $(a+bx)^\alpha$, either normalize first or use $T_{r+1}=a^\alpha\binom{\alpha}{r}\left(\frac{b}{a}x\right)^r$.

When multiplying two series, a requested coefficient may receive contributions from several pairs of terms. List all exponent pairs that add to the required power before summing.

When $\alpha$ is not a non-negative integer, the extended binomial series has the standard convergence condition $|x|<1$. At $x=1$ or $x=-1$, endpoint behavior requires separate analysis and must not be assumed from the open interval.

After substituting $u=g(x)$, apply the condition to the complete inner expression: $|g(x)|<1$. For $(1-3x)^{\frac{1}{2}}$, this gives $|-3x|<1$, hence $|x|<\frac13$.

A non-negative integer exponent produces a polynomial, so its finite expansion is valid for every real x; the convergence restriction is only relevant to the genuinely infinite expansion.

Truncating a convergent series gives an approximation. The closer the inner value is to zero, the more rapidly successive powers usually shrink. Report an approximate sign and appropriate accuracy.

A formula may define a real function on a wider domain than its expansion is valid. For example, the function may exist at a point where this particular power series does not converge.

Scientific calculators and GDCs usually provide factorial, nPr, and nCr commands in a probability or mathematics menu. Enter n first, select the operation, then enter r.

Use technology to evaluate large exact counts, but write the model first. An unexplained calculator output does not show whether order matters or whether restrictions were applied correctly.

For generalized coefficients with negative or fractional $\alpha$, many nCr commands are unavailable. Evaluate the descending product $\frac{\alpha(\alpha-1)\cdots(\alpha-r+1)}{r!}$, or define it as a sequence on the GDC.

A CAS may display a finite expansion or a truncated series, but you must still state the order term and validity condition. Verify signs and coefficients against the first-term pattern.

For approximations, keep full calculator precision through the retained sum and round only the final answer. Compare the size of the last retained term with the required accuracy.

Begin by classifying the problem: staged choices, ordered selection, unordered selection, repeated objects, restricted cases, finite binomial expansion, or infinite generalized expansion.

For counting, define what one outcome means. Then state whether order matters, whether replacement is allowed, and whether cases overlap. A correct formula applied to the wrong model earns little credit.

For an extended expansion, normalize the binomial, identify alpha and the complete inner variable u, write the general term, generate only the required powers, and state |u| < 1.

Use exact values during algebra. Fractions reveal coefficient structure and avoid rounding error. Convert to decimals only when an approximation or contextual answer is requested.

Finish with a reasonableness check: a count must be a non-negative integer no larger than the unrestricted total, and a truncated series should have decreasing term sizes near its center.

Attempt each question before revealing the steps. The set moves from direct counting and notation to restrictions, generalized coefficients, convergence, and approximation.

In an examination, write the structural expression before using technology and state exact answers unless an approximation is requested.

Counting starts with structure: multiply successive compatible stages, add mutually exclusive cases, and decide whether order matters.

Use ${}_{n}P_r$ for ordered selections and $\binom{n}{r}$ for unordered selections. Divide by repeated factorials only when objects are identical.

For a non-negative integer exponent, use the finite theorem. For negative or non-integer $\alpha$, normalize to $(1+u)^\alpha$, use generalized coefficients, and state $|u|<1$.

A strong IB solution labels restrictions and cases, preserves exact arithmetic, shows the general term when extracting a coefficient, and checks that the result is reasonable.