Kate and Danny each have $\$10.$ Together, they flip a fair coin $5$ times. Every time the coin lands on heads, Kate

Kate and Danny each have $\$10.$ Together, they flip a fair coin $5$ times. Every time the coin lands on heads, Kate gives Danny $\$1.$ Every time the coin lands on tails, Danny gives Kate $\$1.$ After the five-coin flips, what is the probability that Kate has more than $\$10$ but less than $\$15?$

(A) $\dfrac5{16}$

(B) $\dfrac12$

(C) $\dfrac{12}{30}$

(D) $\dfrac{15}{32}$

(E) $\dfrac38$

Answer: D

Source: Manhattan GMAT

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