From a bunch of flowers containing 16 red roses and 14 white roses (30 roses in total), four flowers have to be selected. In how many different ways can this selection be made if at least one of the selected flowers is a red rose?

Introduction / Context:
This problem asks you to choose 4 flowers from a mixture of red and white roses, under the condition that at least one of the chosen flowers must be red. It is a classic example of using combinations with a simple condition and is best approached using the complement method: count all possible selections and then subtract the selections that violate the condition (that is, selections with no red roses at all). This method simplifies the counting compared to enumerating many separate subcases.

Given Data / Assumptions:
- Number of red roses: 16.

- Number of white roses: 14.

- Total roses: 16 + 14 = 30.

- We must select exactly 4 flowers.

- Condition: at least one selected flower must be a red rose.

- Flowers of the same colour are distinct in counting, and order of selection does not matter; only the chosen subset counts.

Concept / Approach:
Let us first count the total number of ways to select any 4 flowers from the 30 available roses, without any colour restriction. Then we count how many of those selections contain no red rose (that is, all 4 flowers are white). Selections that meet the "at least one red" condition are all selections minus those with zero red roses. This is the complement approach and is often much easier than splitting into multiple red-white combinations for 1 red, 2 reds, 3 reds and 4 reds.

Step-by-Step Solution:
Step 1: Compute the total number of ways to choose 4 flowers from 30 roses.Step 2: Total selections (no restriction) = 30C4.Step 3: Compute 30C4 = (30 * 29 * 28 * 27) / (4 * 3 * 2 * 1) = 27405.Step 4: Next, count the number of selections with no red roses, that is, all 4 flowers are white.Step 5: There are 14 white roses, so such selections are counted by 14C4.Step 6: Compute 14C4 = (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1) = 1001.Step 7: Selections with at least one red rose = total selections - selections with zero red roses.Step 8: Therefore, required count = 30C4 - 14C4 = 27405 - 1001 = 26404.

Verification / Alternative check:
As a check, we can add up the cases by number of red roses directly. Let r be the number of reds selected (r ranges from 1 to 4). The count becomes sum over r of 16Cr * 14C(4-r). Computing explicitly: for r = 1, we have 16C1 * 14C3 = 16 * 364 = 5824; for r = 2, 16C2 * 14C2 = 120 * 91 = 10920; for r = 3, 16C3 * 14C1 = 560 * 14 = 7840; for r = 4, 16C4 * 14C0 = 1820 * 1 = 1820. Adding these: 5824 + 10920 + 7840 + 1820 = 26404, which matches the complement method result.

Why Other Options Are Wrong:
- 27405 counts all selections with no colour restriction, including those with zero red roses, so it does not enforce the "at least one red" condition.

- 26584 and 26585 are near the correct value but arise from miscalculating one of the combinations or making an arithmetic slip in subtraction.

Common Pitfalls:
Many learners attempt to split the problem into many cases (exactly 1 red, 2 reds, 3 reds, 4 reds) and forget one of the cases or miscalculate a term. Others misinterpret "at least one red" as "exactly one red". The complement method is safer and usually quicker: total minus "no red". Additionally, errors often occur in computing large combinations such as 30C4; doing the multiplication and division carefully or cancelling common factors early reduces mistakes.

Final Answer:
The number of ways to select 4 flowers so that at least one is a red rose is 26404.