Mixed PnC and Probability for Bank Exams

The occasional harder question blends permutation-combination with probability: you count favourable selections with nCr, count total selections the same way, and divide. Because SBI Clerk Prelims sets these rarely, your goal is not mastery but a reliable method for the one selection-style question that might appear — and the judgement to skip it if it turns long. This closing topic ties the chapter together and shows how counting and probability meet in a single practical framework.

Selection Problems Using nCr

A typical bank question gives a bag or a group — say a box of 5 red and 3 blue balls — and asks the probability of drawing a certain colour combination. The total ways to draw the required number is nCr on the whole group, and the favourable ways is a product of nCr counts on each colour. Drawing 2 balls from these 8 has 8C2 = 28 total ways; drawing 2 red has 5C2 = 10 favourable ways, so the probability is 10 over 28, which simplifies to five-fourteenths. Count, count, divide.

At-Least and At-Most Questions

Questions asking for at least or at most a number of items are where candidates lose time. At least one means one or more, so the fastest route is usually the complement: total minus the none case. At most two means zero, one or two, which you may add directly or, again, reach by subtracting the unwanted cases from the whole. The guiding principle is to count whichever side — the wanted cases or the unwanted ones — has fewer terms, then adjust. Choosing the shorter count is the entire speed trick here.

A Fast, Safe Approach

Treat every mixed question as three mechanical steps: find the total number of ways, find the favourable number of ways, then divide and simplify. Do the counting with the small-r nCr shortcuts from the earlier topic, and use the complement whenever the phrase at least one appears. If either count needs more than a couple of nCr terms, the question is eating time and, in a paper with no sectional cut-off, is a sound skip. Speed and selectivity beat completeness in this section.

P = (favourable nCr product) / (total nCr) At least one: 1 − P(none)

Solved Examples

Example 1: From 4 men and 3 women, a committee of 2 is chosen; P(both women) = 3C2 / 7C2 = 3/21 = 1/7. Example 2: P(at least one woman) = 1 − P(no woman) = 1 − (4C2 / 7C2) = 1 − 6/21 = 5/7 — the complement avoids adding the one-woman and two-women cases.

✗ Adding every 'at least one' case one by one  |  ✓ Subtracting the single 'none' case from 1

Honest Weightage and Exam-Day Priority

It is worth being blunt about how little this chapter's PnC and probability portion is worth in SBI Clerk Prelims. Across the 35-question Numerical section, mensuration might supply 1 to 3 direct questions while permutation, combination and probability together rarely exceed a single question, and some slots have none at all. That reality should shape your preparation: memorise the mensuration formulas thoroughly because they are quick, certain marks, but treat PnC and probability as light revision, learning only the order test, the base sample spaces, the small-r nCr shortcuts and the complement rule. On exam day, attempt these questions only when they are visibly one-step; the moment a selection or probability problem shows several cases or large numbers, leave it and bank the guaranteed arithmetic and DI marks instead. With no sectional cut-off and a fierce time limit, this disciplined, honest allocation of effort protects your overall score far better than chasing every last counting question.