A record-route (RR) option is used to record the Internet routers that handle the datagram. It is listed in OPTIONS of $\textsf{IPv4}.$
According to $\textsf{RFC 791},$ there are two cases for the format of an option:
- Case $1:$ A single octet of option-type.
- Case $2:$ An option-type octet, an option-length octet, and the actual option-data octets.
In both the cases, first $16$ bits of OPTIONS field is used. Therefore, out off $40$ Bytes only $38$ Bytes are remaining for storing $\textsf{IPv4}$ addresses. In $38$ Bytes we can store $9\;\textsf{ IPv4}$ addresses as each $\textsf{IPv4}$ address is of $4$ Bytes.
$\therefore 9$ should be answer.
Reference: https://tools.ietf.org/html/rfc791