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stars of switched at birth - Redraw

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A community health researcher in Kansas tracks vaccine distribution across 7 clinics, each receiving either 0 or 1 dose per day. How many 7-day schedules are possible such that no two consecutive days have all 7 clinics administering doses?

📌 Let $s_n$ be the number of valid $n$-day sequences. We define two recurrence states:

📌 But actually, $A_n$ and $B_n$ are counts of entire $n$-length sequences, not per day.

📌 Clarify: each day has $128$ states. But we define recurrence based on type.

📌 Let $t_n$ be the total number of valid $n$-day schedules.

📌 To form $x_n$: any valid sequence of $n-1$ days, followed by a non-all-1s day. There are $127$ such (since $128 - 1 = 127$), and for each, $127$ choices:

📌 But we can keep in terms of recurrence.

📌 Sum: $211,320,000 + 42,264,000 = 253,584,000 + 14,792,400 = 268,376,400$

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