In how many different ways can 100 distinct soldiers be divided into 4 squads of sizes 10, 20, 30 and 40 respectively?

Introduction / Context:
This question is about partitioning a large set of distinct objects into groups of specified sizes. It is a classical combinatorics problem involving multinomial coefficients, where you divide one big group into several labelled subgroups with fixed sizes.

Given Data / Assumptions:

• There are 100 distinct soldiers.
• They must be divided into 4 squads.
• The squad sizes are fixed at 10, 20, 30 and 40 soldiers.
• Squads are considered distinct (first squad of 10, second of 20, and so on).

Concept / Approach:
To divide 100 distinct soldiers into four labelled groups of sizes 10, 20, 30 and 40, we use a multinomial coefficient. One approach is to choose the first squad of 10 soldiers in 100C10 ways, then from the remaining 90 soldiers choose the next 20 in 90C20 ways, then from the remaining 70 choose the next 30 in 70C30 ways, and the final 40 soldiers automatically form the last squad. The total number of ways is the product of these combination counts, which can be compactly expressed using factorials.

Step-by-Step Solution:
First choose the squad of 10: 100C10 ways.Then choose the squad of 20 from the remaining 90: 90C20 ways.Next choose the squad of 30 from the remaining 70: 70C30 ways.The remaining 40 soldiers automatically form the last squad.Total ways = 100C10 * 90C20 * 70C30.In factorial form, this simplifies to 100! / (10! * 20! * 30! * 40!).This value is extremely large and does not equal any of the small numerical options listed.

Verification / Alternative check:
The structure matches the general formula for splitting n distinct objects into groups of sizes n1, n2, n3 and n4: n! / (n1! * n2! * n3! * n4!).For n = 100 and group sizes 10, 20, 30, 40, the formula gives 100! / (10! * 20! * 30! * 40!).

Why Other Options Are Wrong:
1700 and 190 are tiny compared to the true count and ignore the factorial growth inherent in the problem.18! does not reflect the correct group sizes and is unrelated to the actual splitting of 100 soldiers.Therefore, none of the given numeric values is correct.

Common Pitfalls:
Trying to treat squads as indistinguishable, which would require dividing further by 4!.Using an incorrect formula such as simply 4^100, which counts assignments without controlling squad sizes.Not recognizing that the correct answer is best left in factorial form and hence choosing an incorrect small option.

Final Answer:
The correct count is 100! / (10! * 20! * 30! * 40!), so the appropriate option is “None of these”.