Permutations Combinations Binomial Questions for AGNIVEER NAVY

Factorials and Permutations (nPr)

A factorial (n!) is the product of all positive integers up to n. It counts ordered arrangements. Permutation nPr counts the number of ways to arrange r objects chosen from n distinct objects, where ORDER MATTERS. Think of it as filling r slots with n choices, where each choice reduces available options. For example, selecting a captain and vice-captain from 5 people is different from selecting 2 people for a committee (order matters in the first case, not the second).

• n! = n × (n-1) × (n-2) × ... × 1; by definition 0! = 1
• nPr = n! / (n-r)! — counts ordered selections
• nPr is always larger than nCr for the same n and r
• Circular permutations: (n-1)! because rotations are identical
• Permutations with repetition: n^r when objects can repeat

Combinations (nCr)

A combination nCr counts the number of ways to select r objects from n distinct objects where ORDER DOES NOT MATTER. Use nCr for committees, teams, groups, or any selection where arrangement is irrelevant. The key insight: nCr = nPr / r! because we divide out the r! internal arrangements that don't matter. Combinations are always smaller than permutations.

• nCr = n! / (r!(n-r)!) — counts unordered selections
• nCr = nC(n-r) — symmetry property (choosing 2 from 5 = choosing 3 to leave out)
• nC0 = nCn = 1 (one way to choose nothing or everything)
• nC1 = n (one way to choose each individual item)
• Use combinations for committees, groups, teams, subsets

Binomial Theorem & Coefficients

The binomial theorem expands (a+b)^n into a sum of terms with binomial coefficients nCr. Each term has the form nCr × a^(n-r) × b^r. The coefficients nC0, nC1, nC2, ..., nCn form Pascal's triangle. In Agniveer exams, you'll typically find the coefficient of a specific term or use properties like sum of all coefficients = 2^n. The middle term is important for even n.

• (a+b)^n = sum of nCr × a^(n-r) × b^r for r = 0 to n
• General term (r+1)th = nCr × a^(n-r) × b^r
• Sum of all binomial coefficients: nC0 + nC1 + ... + nCn = 2^n
• For (1+x)^n, coefficient of x^r is simply nCr
• Middle term: when n is even, it's the (n/2 + 1)th term

Distinguishing nPr vs nCr in Word Problems

The most common mistake is confusing when to use permutations vs combinations. The deciding question: Does the order or arrangement matter? If you're selecting people for different roles (captain, vice-captain, treasurer), use nPr. If you're just forming a group with no roles, use nCr. In Agniveer exams, word problems often hide this distinction. Read carefully for words like 'arrange', 'select', 'form', 'choose'.

• Use nPr: arranging, ordering, seating, assigning different roles, passwords, codes
• Use nCr: selecting, choosing, forming committees, picking teams, grouping
• Key phrase test: 'In how many ways can we arrange?' → nPr; 'In how many ways can we select?' → nCr
• Combination problems often involve committees, teams, or subsets with no internal hierarchy
• Permutation problems involve ranks, positions, or sequences where position matters

• Confusing nPr and nCr: Using nPr when nCr is needed (or vice versa). Always ask: does order matter? If no, use nCr.
• Forgetting that 0! = 1: This causes errors in nP0 and nC0 calculations. Remember: there's exactly 1 way to arrange nothing.
• Misidentifying the general term in binomial: Students often forget the (r+1)th term uses nCr, not nCr+1. The index r starts at 0.
• Not simplifying nCr before multiplying: Writing 8!/(3!×5!) and then calculating 8! fully instead of canceling: (8×7×6)/(3×2×1).
• Ignoring the symmetry property nCr = nC(n-r): This can save time. For example, 10C8 = 10C2, which is much faster to calculate.
• Misreading circular arrangement as linear: Circular permutations use (n-1)!, not n!. This is frequently tested in Agniveer.

• P = Position matters (Permutation); C = Committee (Combination) — no positions, just selection
• nPr > nCr always — Permutations count more because order creates more distinct arrangements
• Circular = (n-1)! — One person sits, then arrange the rest in a line around the circle
• Binomial coefficients = Pascal's triangle — Each row sums to 2^n; use symmetry nCr = nC(n-r) to halve your work
• General term index: T(r+1) uses nCr, not nC(r+1) — The term number is r+1, but the coefficient index is r

• Agniveer Navy typically asks 1 direct formula question (calculate nPr or nCr) and 1-2 word problems. The word problems test your ability to identify whether order matters.
• Circular arrangement questions appear frequently in Agniveer SSR/MR papers. Always use (n-1)! and mention why rotations are identical.
• Binomial questions usually ask for a specific term's coefficient or the sum of coefficients. Memorize that sum = 2^n for (a+b)^n.
• Time-saving tip: Use the symmetry property nCr = nC(n-r). If r > n/2, calculate nC(n-r) instead. For example, 15C12 = 15C3 = 455 (much faster).
• Recent Agniveer papers show preference for multi-step problems: 'From 12 people, form a committee of 5 with at least 2 women.' These require breaking into cases and adding nCr values. Practice case-based problems.