Question 1177694
Let's solve each part of the problem step-by-step.

**Understanding Combinations**

We'll use combinations (nCr) to calculate the number of ways to choose routines. The formula for combinations is:

* nCr = n! / (r! * (n-r)!)

Where:

* n = total number of items
* r = number of items to choose

**A. 4 Moderately Difficult Routines**

* Total moderately difficult routines: 5
* Routines to choose: 4

* Number of ways = 5C4 = 5! / (4! * (5-4)!) = 5! / (4! * 1!) = 5

**B. 4 Easy or Moderately Difficult Routines**

* Total easy routines: 2
* Total moderately difficult routines: 5
* Total easy or moderately difficult routines: 2 + 5 = 7
* Routines to choose: 4

* Number of ways = 7C4 = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35

**C. 2 Moderately Difficult and 2 Difficult Routines**

* Moderately difficult routines: 5
* Difficult routines: 3

1.  Choose 2 moderately difficult routines: 5C2 = 5! / (2! * 3!) = (5 * 4) / (2 * 1) = 10
2.  Choose 2 difficult routines: 3C2 = 3! / (2! * 1!) = 3

* Total number of ways = 10 * 3 = 30

**D. 1 Easy and 3 Difficult Routines**

* Easy routines: 2
* Difficult routines: 3

1.  Choose 1 easy routine: 2C1 = 2! / (1! * 1!) = 2
2.  Choose 3 difficult routines: 3C3 = 3! / (3! * 0!) = 1

* Total number of ways = 2 * 1 = 2

**Answers**

* **A. 4 moderately difficult routines:** 5 ways
* **B. 4 easy or moderately difficult routines:** 35 ways
* **C. 2 moderately difficult and 2 difficult routines:** 30 ways
* **D. 1 easy and 3 difficult routines:** 2 ways