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Sig·Software Engineer·Technical Phone Screen·Junior

Junior
May 2026

Summary

SIG quant researcher interview with a combinatorics problem that felt deceptively clean at first glance. The cookie arrangement question is the kind of thing that sounds like a five-second answer until you realize you're mixing two constraints at once.

Questions Asked (1)

Q1

You have 4 indistinguishable snickerdoodle cookies and 7 indistinguishable chocolate-chip cookies, 11 total. How many distinct arrangements exist if you line up exactly 5 of them?

Algorithms & Data Structures
Author's notes

I jumped straight to 11 choose 5 and then realized that's wrong because the cookies of the same type are identical.

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AI HintsAI Generated

Suggested Approach

Recognize that the cookies are indistinguishable within their types, so the problem reduces to counting the number of ways to choose 5 cookies from 4 snickerdoodle and 7 chocolate-chip cookies. Enumerate the possible numbers of snickerdoodle cookies (0 to 4) and for each, check if the remaining chocolate-chip cookies needed (5 - s) is available (≤7). Sum the valid combinations.

Pro tip: Clarify with the interviewer whether the cookies of the same type are truly indistinguishable and whether the order of the 5 cookies matters. This shows attention to detail and avoids misinterpretation.

1. Understand the problem

Confirm that cookies of the same type are indistinguishable and that we are counting distinct sequences of 5 cookies (order matters).

2. Define variables

Let s be the number of snickerdoodle cookies in the arrangement, where s can range from 0 to 4 (since only 4 are available).

3. Determine constraints

For each s, the number of chocolate-chip cookies needed is 5 - s. This must be between 0 and 7 (since 7 are available).

4. Enumerate valid s

Check each s from 0 to 4: s=0 requires 5 chocolate-chip (valid), s=1 requires 4 (valid), s=2 requires 3 (valid), s=3 requires 2 (valid), s=4 requires 1 (valid). All are valid.

5. Count arrangements

For each valid s, the number of distinct sequences is C(5, s) (choose positions for the snickerdoodle cookies). Sum these: C(5,0)+C(5,1)+C(5,2)+C(5,3)+C(5,4) = 1+5+10+10+5 = 31.

Key Points to Mention

  • Indistinguishable items within each type mean we only care about counts, not identities.
  • The problem is equivalent to counting binary strings of length 5 with at most 4 zeros (snickerdoodle) and at most 7 ones (chocolate-chip).
  • Use combinations to count arrangements for each possible count of snickerdoodle cookies.
  • The sum of binomial coefficients C(5,0) through C(5,4) equals 2^5 - C(5,5) = 31.
  • Alternatively, use generating functions: coefficient of x^5 in (1+x+...+x^4)(1+x+...+x^7).
  • Edge cases: ensure not to exceed available counts; here all s from 0 to 4 are valid because 5-s ≤ 7 for all.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.